Browse Sequences & Series problems Studdy Solved!

Problem 301

Find a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence $3, -3, -9, \ldots$

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Problem 302

Find a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 23, 27, 31, ...

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Problem 303

Find the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

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Problem 304

Find the sums of these sequences: a. $1+2+3+\ldots+999$ b. $1+3+5+\ldots+1003$ c. $4+9+14+\ldots+489$ d. $293+290+287+\ldots+2$ a. The sum is:

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Problem 305

Find the sums of these sequences using Gauss's method: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

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Problem 306

Find the 100th and nth term for these sequences: a. $1, 5, 9, 13, \ldots$ b. $80, 110, 140, \ldots$ c. $1, 2, 4, \ldots$ d. $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ e. $198 + 5 \cdot 2^{31}, 198 + 6 \cdot 2^{31}, 198 + 7 \cdot 2^{31}, \ldots$

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Problem 307

Find the 100th term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ in exponential notation.

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Problem 308

Find the 100th term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ and express it in exponential form.

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Problem 309

Find the $n$ th term of the sequence: $198 + (n+4) \cdot 2^{31}$ .

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Problem 310

Find the nth term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ and simplify your answer.

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Problem 311

Find the relationship between consecutive numbers in the sequence: $2, 6, 18, 54, 162, \ldots$

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Problem 312

Find the relationship between consecutive numbers in the sequence: $4, 12, 36, 108, 324, \ldots$

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Problem 313

Shannon has job offers: Company ABC: \$45,500 with 8\% raise yearly; Company XYZ: \$62,000 with 3\% raise. After 15 years, which is better?

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Problem 314

Malik has a \ $50 gift card for movies costing \$6 each. Write a sequence rule and find how many movies he can see.$ $f(x)=$ $ Malik can see movies to deplete the card.

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Problem 315

A cab driver charges \ $4 per mile. The meter shows \$7.50 for 2 miles, \$11 for 3 miles. Find fares for 4, 5, 6 miles.$ $f(4)=$ $f(5)=$ $f(6)=$ $ Also, write a recursive rule for this fare structure.

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Problem 316

A cab charges \ $4 per mile, with an arithmetic fare. Find fares for 4, 5, and 6 miles:$ \mathrm{f}(4) $,$ \mathrm{f}(5) $,$ \mathrm{f}(6)$. Write a recursive rule.

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Problem 317

What comes next in the series: $a 3 Z, b 6 Y, c 9 X, d 12 W, e 15 V, f 18 U$ ? Options: A. g21T B. $924 \mathrm{U}$ C. $\mathrm{G} 24 \mathrm{~T}$ D. $g 21 t$ E. $\mathrm{f} 21 \mathrm{~T}$ F. $g 24 t$

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Problem 318

Find the sixth term in the sequence starting with 5, where each term is $2 \times \text{previous} - 4$ .

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Problem 319

Write the sum using sigma notation: $6-8+10-12+14-16+18=\sum_{n=1}^{A}$ , where $\begin{array}{l} A= \\ B= \end{array}$

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Problem 320

Find the 82nd term of the arithmetic sequence $-10,6,22, \ldots$

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Problem 321

How many terms are in this series? * $\sum_{n=2}^{9} 18(0.1)^{n}$

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Problem 322

Determine la siguiente sumatoria. Escriba el resultado redondeado al entero más cercano. Sólo escriba dígitos en la contestación. No escriba comas ni puntos. $\sum_{i=1}^{100}\left(\frac{i^{3}}{3}-2 i^{2}\right)$
Add your answer Integer, decimar, or Enotation allowed

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Problem 323

Consider the series $\sum_{n=1}^{\infty} \frac{1}{n(n+4)}$
Determine whether the series converges, and if it converges, determine its value. Converges $(\mathrm{y} / \mathrm{n}$ ): $\square$ Value if convergent (blank otherwise): $\square$
Note: You can earn partial credit on this problem.

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Problem 324

Time left 0:06:46 Hide
By using proof by induction specify the firbe arrec sieps iv prove that $P(n)$ is true such that $P(n)$ is: $8+16+24+\cdots+8 n=\frac{8 n(n+1)}{2}$ is true for all positive integers $(n \geq 1)$

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Problem 325

Identify the incorrect expression for the sum $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ from the options given.

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Problem 326

Write an equation for the sequence $7, 9, 11, 13, 15, \ldots$ and find the 20th term.

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Problem 327

Find an equation for the sequence $7, 9, 11, 13, 15, \ldots$ and determine the 20th term.

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Problem 328

Find the explicit formula for the sequence: 15, 19, 23, 27, ...

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Problem 329

Find the recursive formula for the sequence: 15, 19, 23, 27, ...

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Problem 330

Find the recursive formula for the sequence where $a_{1}=-1$ and $d=-10$ .

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Problem 331

Find the explicit formulas for the sequences with:
9. $a_1 = -1$ , $d = -10$
10. $a_1 = -3$ , $d = -9$

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Problem 332

Find the explicit formula for the sequence with $a_{1}=-1$ and common difference $d=-10$ .

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Problem 333

James deposits \ $1440.71 each quarter into an annuity account for his child's college fund in order to accumulate a future value of$ \ $95,000$ in 13 years. How much of the $\$ 95,000$ will James ultimately deposit in the account, and how much is interest earned? Round your answers to the nearest cent, if necessary.
Formulas

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Problem 334

3. [15] a) Find the $5^{\text {th }}$ degree Taylor polynomial of $8 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12$ about $x=0$ and [6] about $x=20$ , then simplify each result. What do you observe? b) Find the $100^{\text {th }}$ degree Taylor polynomial of $8 x^{5}-6 x^{4}+3 x^{3}+27 x^{2}+32 x-12$ about $x=0$ [6] and about $x=20$ , then simplify each result. What do you observe? c) What can you conclude about the $n^{\text {th }}$ degree Taylor polynomial about $x=a$ of a polynomial of [3] degree $m$ , where $n \geq m$ ?

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Problem 335

20. Seorang pemilik modal menanamkan uangnya di sebuah bank dengan bunga $12 \%$ per tahun. Bunga serta modal ditanamkan kembali pada bank tersebut dengan bunga $15 \%$ per tahun. Kemudian, bunga dan modal tersebut ditanamkan kembali dengan bunga $13 \%$ per tahun. Berapakah rata-tata tingkat bunga yang diperoleh orang tersebut?

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Problem 336

(a) Verify that for all $n \geq 1$ , $2 \cdot 6 \cdot 10 \cdot 14 \cdots \cdots(4 n-2)=\frac{(2 n)!}{n!}$ (b) Use part (a) to obtain the inequality $2^{n}(n!)^{2} \leq(2 n)$ ! for all $n \geq 1$ .

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Problem 337

The $n^{\text {th }}$ term of a sequence is given by $2 n+5$ Work out the first three terms of this sequence. a) How much does the sequence go up by between each term? b) Explain how you could have used the $n^{\text {th }}$ term rule to answer part a) without working out any of the terms.

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Problem 338

The $n^{\text {th }}$ term rule for a sequence is $T(n)=n^{2}+4$ Work out the $5^{\text {th }}$ term of this sequence.

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Problem 339

The $n^{\text {th }}$ term rule for a sequence is $T(n)=n^{2}+3$
Work out the $5^{\text {th }}$ term of this sequence.

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Problem 340

Taylor Series Formula $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)(x-a)^{n}}{n!}$
Find the Taylor Series for the following functions at the given value by using the definition and finding the derivatives at the given value.
1. $f(x)=e^{x}$ at $a=e$
4. $f(x)=\cos (2 x)$ at $a=\frac{\pi}{4}$
2. $f(x)=e^{2 x}$ at $a=3$
5. $f(x)=\frac{1}{x} \quad$ at $\quad a=7$
3. $f(x)=\cos (x)$ at $a=-\pi$
6. $f(x)=\ln (x)$ at $a=e$

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Problem 341

8. Write the explicit and recursive formula for the sequence $-4,-1,2, \ldots$

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Problem 342

Write out the first three terms and the last term. Then use the formula for the sum of the first $n$ terms of an arithmetic sequence to find the indicated sum. $\sum_{i=1}^{25}(9 i-2)$

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Problem 343

11. 1) 10] Consider the infinite series: $\sum_{n=1}^{\infty} a_{n}$
With the partial sums, $S_{n}$ . a) If $\lim _{n \rightarrow \infty} a_{n}=\sqrt{\pi}$ , what can we say about the series convergence or divergence? b) If $\lim _{n \rightarrow \infty} S_{n}=\sqrt{\pi}$ , what can we say about the series convergence or divergence?

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Problem 344

Consider the following series. $\sum_{n=1}^{\infty} \frac{3 n^{2}-1}{4 n^{5}+3 n+1}$
Use the Limit Comparison Test to complete the limit.

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Problem 345

Determine whether the following series converges or diverges. $\sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{1}{n}\right)$
Input $C$ for convergence and $D$ for divergence: $\square$
Note: You have only one chance to enter your answer.

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Problem 346

(1 point)
Classify each series as absolutely Convergent, Conditionally Convergent, or Divergent. (a) $\sum_{k=1}^{\infty} \frac{(-7)^{k}}{k^{2}}$ ? (b) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!}$ ?

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Problem 347

Étudiez la convergence (absolue ou conditionnelle) de la série alternée : $\sum_{n=1}^{\infty}(-1)^{n}\left(\frac{n}{7 n^{2}+3}\right)$

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Problem 348

You have \$400,000 saved at 4\% interest. How much can you withdraw monthly for 20 years?

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Problem 349

Find the formula for the $n^{\text{th}}$ term of the sequence $5, 9, 13, 17, 21, \ldots$ . What is $a_{n}=$ ?

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Problem 350

Find the 10th term in the arithmetic sequence $11, 19, 27, 35, 43, \ldots$ . Options: $91, 75, 100, 83$ .

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Problem 351

Find the max price per share for Grips Tool if growth is 25% for 3 years, then 10%, and required return is 15%.

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Problem 352

Find the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

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Problem 353

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

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Problem 354

Marika starts sprinting 100 yards and adds 4 yards daily for 21 days. Find her distance on day 21 using $a_{n}=100+(n-1)4$ . Options: A. 200 B. 184 C. 221 D. 180.

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Problem 355

Find the common ratio of the geometric sequence where $\mathrm{a}_{1}=1$ and $\mathrm{a}_{9}=25$ .

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Problem 356

수렴하는 수열을 고르세요: ᄀ. $\left\{\frac{1}{2 n+1}\right\}$ , ㄴ. $\left\{\frac{(-1)^{n}}{n}\right\}$ , ᄃ. $\left\{(-1)^{n}+2\right\}$ .

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Problem 357

Calculate the stock value of Delphi Products with a current dividend of \$2, growing at varying rates and an 18% return.

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Problem 358

Find the pattern in the sequence $-2, -8, -14, -20, \ldots$ .

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Problem 359

$\sum_{k=1}^{n}\left(b_{k}-9 a_{k}\right)$ where $a_k = 7$ and $b_k = 30$ .

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Problem 360

Write the sum using sigma notation: $7+9+11+13+\ldots+23=\sum_{n=1}^{A} B$ , where $A=$ $\square$ $B=$ $\square$

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Problem 361

Find the sum of the series $S=\sum_{i=1}^{\infty} x^{4 i},|x|<1$ $S=$

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Problem 362

A rumour is being spread in a community that the mayor is going to retire before the next election. Initially, 6 people were told the rumour and each of these people told 6 others the following day. Each of the 6 people told by the initial people told 6 others the day after they were told. If this process keeps up, how many people will be told the rumour on the eighth day? a. $\quad 48$ people b. $\quad 279936$ people c. 1679616 people d. 10077696 people

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Problem 363

7 The third term of an arithmetic sequence is twice the first term. The second term is 45 . a) Find the value of the first term and the common difference. b) Determine whether or not 310 is a term of the sequence.

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Problem 364

5. What are the first three terms of the Arithmetic Sequence $a_{n}=11-3(n-1) ?$ $0,8,16$ $11,8,5$ $1,2,3$ $8,5,2$

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Problem 365

PERGUNTAS/ORIENTAÇÖES Leia atentamente a prova e responda com clareza as seguintes questōes
1. Dados conjuntos $A=\left\{x \in R:-3 \leq x \leq \frac{1}{4}\right\}, B=\{x \in R:-1<x \leq 5\} e$ $C=\left\{x \in R:\left|5 x^{2}-6\right| x|-8|-\left|3 x^{2}-12\right| x|+12| \leq 0\right\}$ , determina os intervalos e as seguintes operaçōes: a) $A \cap B$ b) $\mathrm{B} \cap \mathrm{C}$ c) $C-A$ d) $A \cap C$
2. Ache o campo de existências das funçōes. a) $f(x)=\frac{1}{x^{2}-36}$ b) $y=\sqrt{5+x}-\sqrt{5-x}+3$ c) $y=\log _{(x-3)}\left(6 x-5-x^{2}\right)$
3. Dadas as sucessões numérica, determine o termo geral das sucessões e vigésimo termo. a) $\frac{4}{3}, \frac{7}{8}, \frac{10}{15}, \frac{13}{24}, \ldots$ b) $1,3,5,7 \ldots$
4. Calcule os seguintes limites. a) $\lim _{x \rightarrow \infty}\left(\frac{1+3+5+7+\cdots+(2 x-1)}{x+1}-\frac{2 x+1}{2}\right)$ b) $\lim _{x \rightarrow \infty} \frac{\sqrt{x}}{\sqrt{x+\sqrt{x+\sqrt{x}}}}$ c) $\lim _{x \rightarrow \infty} \frac{x^{2}-3 x-4}{\sqrt{x^{4}+x-2}}$
Pensamento: A velocidade de um veiculo não altera a distância, mas sim diminui o tempo viagem.
Cotação: 1-a) 1 valor; 1-b) 2 valores; 1-c) 2 valores; 1-d) 2 valores; 2 - 5 valores; 3 - 3,5 valores; 4-4,5 valores

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Problem 366

(a) Use differentiation to find a power series representation for $\begin{array}{l} f(x)=\frac{1}{(9+x)^{2}} \\ f(x)= \sum_{n=0}^{\infty}( \end{array}$
What is the radius of convergence, $R$ ? $R=$ (b) Use part (a) to find a power series for $\begin{array}{c} f(x)=\frac{1}{(9+x)^{3}} \\ f(x)=\sum_{n=0}^{\infty}(\square) \end{array}$
What is the radius of convergence, $R$ ? $R=$ (c) Use part (b) to find a power series for $\begin{array}{c} f(x)=\frac{x^{2}}{(9+x)^{3}} \\ f(x)=\sum_{n=2}^{\infty}(\square) \end{array}$
What is the radius of convergence, $R$ ? $R=$ $\square$

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Problem 367

1 Earlier, we learned that the $n$ th term of a geometric sequence with an initial value of $a$ and a common ratio of $r$ is $a\left(r^{n-1}\right)$ . For a Koch Snowflake, it turns out that we can find the number of triangles added on at each iteration by having $a=3$ and $r=4$ . The sum $s$ of the first $n$ terms in this geometric sequence tell us how many triangles total make up the $n$ th iteration of the snowflake $s=3+3(4)+3\left(4^{2}\right)+\ldots+3\left(4^{n-1}\right)$
More generally, the sum of the first $n$ terms of any geometric sequence can be expressed as $s=a+a(r)+a\left(r^{2}\right)+\ldots+a\left(r^{n-1}\right)$ or $s=a\left(1+r+r^{2}+\ldots+r^{n-1}\right)$
1. What would happen if we multiplied each side of this equation by $(1-r)$ ? (hint: $(x-1)\left(x^{3}+x^{2}+x+1\right)=x^{4}-1$ .)
Type your answer in the box. $-b(1-r)=a\left(1-r{ }^{r}\right)$
2 2. Rewrite the new equation in the form of $s=$ . Type your answer in the box. $s=a\left(\frac{1-r^{1}}{1-r}\right)$
3. Use this new formula to calculate how many triangles after the original are in the first 5 , 10 , and 15 iterations of the Koch Snowflake.

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Problem 368

Bob makes his first $\$ 900$ deposit into an IRA earning $7.3 \%$ compounded annually on his 24 th birthday and his last $\$ 900$ deposit on his 38th birthday ( 15 equal deposits in all). With no additional deposits, the money in the IRA continues to earn $7.3 \%$ interest compounded annually until Bob retires on his 65th birthday. How much is in the IRA when Bob retires?
The amount in the IRA when Bob retires is $\$$ $\square$ (Round to the nearest cent as needed.)

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Problem 369

Find the sum of the given finite geometric series. $6+\frac{6}{3}+\frac{6}{9}+\frac{6}{27}+\ldots+\frac{6}{59049}$
The sum of the finite geometric series is $\square$ (Type an integer or a simplified fraction.)

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Problem 370

Use the Root Test to determine whether the series convergent or divergent. $\sum_{n=2}^{\infty}\left(\frac{-3 n}{n+1}\right)^{6 n}$
Identify $a_{n}$ . $\left(\frac{-3 n}{n+1}\right)^{6 n}$
Evaluate the following limit. $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$ $\square$
Since $\lim _{n \rightarrow \infty} \sqrt[n]{\left|a_{n}\right|}$
1, the series is divergent
Enhanced Feedback Please try again, keeping in mind that you can use the Root Test. Need Help? Read It

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Problem 371

Find the explicit formula for the sequence $-7, -4, -7, 2, 5$ . Choose from A, B, C, or D.

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Problem 372

Find the 500th term of the sequence $24, 31, 38, 45, 52, \ldots$ using $a_{n}=a_{1}+(n-1) \cdot d$ . A. 3493 B. 3545 C. 3517 D. 3524

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Problem 373

Find the formula for Marika's daily sprint distance after 21 days: A. $a_{n}=100+(n-1) 4$ B. $a_{n}=4+(n-1) 100$ C. $a_{n}=100+(n-1) 21$ D. $a_{n}=21+(n-1) 4$

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Problem 374

Zra sprints 100 yards and adds 5 yards daily for 21 days. Find his distance on day 21 using $a_{n}=100+(n-1)5$ . Options: A. 200 B. 100 C. 225 D. 205

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Problem 375

Find the 47th term of the sequence given by $a_{n}=-11+(n-1)(-3)$ . Options: A. -149 B. -152 C. 127 D. -138

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Problem 376

Find the explicit formula for the sequence $5, 2, -1, -4, \ldots$ . Choose from the options A, B, C, or D.

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Problem 377

Find the 7th term in the sequence given by $a_{n}=-3 \cdot(3)^{(n-1)}$ . Options: A. 6561 B. -6561 C. -2187 D. 2187

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Problem 378

Identify the geometric sequence from the options: A. $3,-15,-33,-51,-69$ B. $2,3,5,9,17$ C. $4,2,1,\frac{1}{2},\frac{1}{4}$ D. $3,6,9,12$

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Problem 379

Find the common ratio of the geometric sequence: $3, 12, 48, 192, \ldots$ A. 9 B. 4 C. 16 D. 3

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Problem 380

Find the common ratio of the geometric sequence: $64, 16, 4, 1, \ldots$ . A. 4 B. $\frac{1}{4}$ C. 8 D. $\frac{1}{8}$

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Problem 381

Find the recursive formula for the sequence $2, -10, 50, -250, \ldots$ . Options: A, B, C, D.

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Problem 382

Find the explicit formula for the number of rabbits in generation $n$ if there are 108 rabbits in generation 3: $a_{n}=6 \cdot 3^{(n-1)}$ .

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Problem 383

Find the formula for rabbit population in the $n$ th generation if it starts with 3 and multiplies by 6 each generation. A. $a_{n}=6 \cdot 3^{(n-1)}$ B. $a_{n}=3 \cdot\left(\frac{1}{6}\right)^{(n-1)}$ C. $a_{n}=6 \cdot\left(\frac{1}{3}\right)^{(n-1)}$ D. $a_{n}=3 \cdot 6^{(n-1)}$

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Problem 384

Find the 8th term of the geometric sequence given by $a_{n}=6 \cdot(-2)^{(n-1)}$ . Options: A. -768 B. 768 C. -1536 D. 1536

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Problem 385

Find the sums of these sequences without formulas: a) $1+2+3+\ldots+69$ b) $1+3+5+\ldots+2009$ c) $6+12+18+\ldots+600$ d) $1000+990+\ldots+10$

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Problem 386

Identify the function for a geometric sequence: A. $A(n)=P+(n-1) i \cdot P$ B. $A(n)=n+(i-1) \cdot P$ C. $A(n)=(n-1)(P \cdot n)^{i}$ D. $A(n)=P(1+i)^{n-1}$

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Problem 387

Find the formula for bacteria in generation $n$ if it starts at 120 and triples each generation: $a(n)=120 \cdot 3^{n-1}$ .

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Problem 388

Miriam owes \ $550 with a monthly compound interest of$ 1.8\%$. Which sequence shows her increasing balance?

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Problem 389

Karri deposits \ $1600 initially and \$300 monthly. Find the formula for her account balance at month$ n$.

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Problem 390

Lesson 6 Checkpoint EVALUATE
Independent Practice \begin{tabular}{|l|c|c|} \hline Learning Goals & Lesson Reflection (circle one) \\ \hline \begin{tabular}{l} I can determine an explicit expression or steps for \\ calculation from a context \end{tabular} & Starting... Getting There... Got it! \\ \hline \end{tabular} Complete the previous problems, check your solutions, then complete the Lesson Checkpoint below. Complete the Lesson Reflection above by circling your current understanding of the Learning Goal(s).
Write an explicit rule in function notation for the arithmetic sequence.
1. A student loan needs to be paid off beginning the first year after graduation. Beginning at Year 1 , there is $\$ 52,000$ remaining to be paid. The graduate makes regular payments of $\$ 8,000$ each year. The graph shows the sequence.
A home cook needs to boil 1 liter of water on the stove. Water boils at $100^{\circ} \mathrm{C}$ . \begin{tabular}{|l|c|c|c|} \hline Minutes & 3 & 4 & 5 \\ \hline Temperature $\left({ }^{\circ} \mathrm{C}\right)$ & 38 & 46 & 54 \\ \hline \end{tabular}
2. What is the common difference of the sequence represented in the table?
3. What is the explicit rule for the sequence?
4. Find the temperature at 10 minutes. Is the water boiling yet? lifelong Algebra 1A (2024) Module 3

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Problem 391

The radioactive isotope ${ }^{226} \mathrm{Ra}$ has a half-life of approximately 1599 years. Consider a lab that currently has 45 g of ${ }^{226} \mathrm{Ra}$ . (a.) How much of the isotope remains after 1200 years? Round your answer to three decimal places.

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Problem 392

MATH102 (Calculus II) First exam - Page 2 of 4 April 25, 2024
1. $\left(21 / 2\right.$ points) The sum of the series $\sum_{k=0}^{\infty} \frac{5^{k} 3^{-k}}{2^{k}}$ is A. 6 B. 5 C. -6 D. -5
2. ( $21 / 2$ points) One of the following values of $p$ makes the series $\sum_{k=0}^{\infty} \frac{p^{k} k^{2}}{k^{3}+1}$ conditionally convergent. A. -1 B. 1 C. 2 D. 3
3. (2 $1 / 2$ points) The series $\sum_{k=0}^{\infty}(-1)^{k} \frac{3}{k^{2}+1}$ is A. conditionally convergent. B. divergent. C. absolutely convergent.
4. ( $21 / 2$ points) Which of the following series is convergent? A. $\sum_{k=1}^{\infty} \frac{k}{k^{2}+4}$ B. $\sum_{k=1}^{\infty} \frac{1}{k+3}$ $\begin{array}{l} \frac{k}{k^{2}+4}<\frac{n}{n^{2}}=\frac{1}{n} \times \frac{n}{n^{2}+4}>\frac{n}{2 n^{1}}=\frac{1}{2 n} \mathrm{dim} \\ \frac{1}{k+3}<\frac{1}{k+3}>\frac{1}{2 k} \text { Jir } \end{array}$ C. $\sum_{k=1}^{\infty} \frac{1}{k\left(1+\ln ^{2}(k)\right)}$ D. $\sum_{k=1}^{\infty}(-1)^{k} \frac{4 k+3}{3 k-5}$
5. ( $21 / 2$ points) One of the following values of $p$ makes the series $\sum_{n=1}^{\infty} \frac{n!}{p^{n}((n+1)!+1)}$ convergent A. 2 B. 0.2 C. 0.5 D. 1 $\begin{aligned} \lim & \frac{(n+1)!}{P^{n+1}((n+1+1)!+1)} \cdot \frac{P^{n}((n+1)!+1)}{n!} \\ & =\frac{(n+1) n!}{P^{2} \cdot P((n+2)(n+1)!+1)} P^{D^{2}((n+1)!+1)}=\ln \frac{(n+1)((n+1)!+1}{P((n+2)(n+1)!} \end{aligned}$ $\begin{array}{l} \lim \frac{(n+1) \cdot((n+1)!+1)}{P((n+2)!+1)}=\lim \frac{(n+1)((n+1)!+1)(n+1)!+(n}{P(n+2)!+P} \\ \left|\frac{1}{P}\right| \lim \frac{(n+1)((n+1)!+1)}{(n+2)!+1)} \quad P((n+2)!+1) \quad P(n+2)!+P \end{array}$

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Problem 393

progresia geometrică $\left(b_{n}\right)_{n \geq 1}$ de ratie $q$ este definita prin anumite elemente date. Determinaţi, in fiecare din cazuri, elementele cerute. (9) $b_{1}=2, q=-3$ . Calculaţi $b_{6} ;{ }^{2} b_{1}=\sqrt{2}, b_{2}=\sqrt{8}$ . Calculaţi $b_{10} ;$ (3) $b_{1}=-\frac{1}{2}, b_{8}=-2^{6}$ . Calculaţi $q$ ; 4) $b_{5}=61, b_{11}=1647$ . Calculaţi $b_{7}$ ; (5) $b_{1}-b_{2}=8, b_{2}+b_{3}=12$ . Calculati $b_{1}, q$ ; 6) $b_{2}+b_{4}=60, b_{1}+b_{3}$ 20. Calculaţi $b_{1}, q$ ; (7) $b_{5}-b_{1}=160, b_{4}-b_{2}=48$ . Calculati $b_{1}, q$ ; (8) $b_{5}+b_{2}-b_{4}=66, b_{6}+b_{3}-b_{5}=-132$ . Calculatil $b_{1}, q$ .

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Problem 394

b) $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{3^{n}}$

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Problem 395

1 Recall that for any geometric sequence starting at $a$ with a common ratio $r$ , the sum $s$ of the first $n$ terms is given by $s=a \frac{1-r^{n}}{1-r}$ . Find the approximate sum of the first 50 terms of each sequence:
Type your answers in the boxes.
1. $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots . s=1$
2. $1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots s=2$

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Problem 396

2) If $b_{0}=1.4, b_{1}=8$ and $b_{m}=\frac{3-b_{m-2}}{b_{m-1}}$ , then $b_{3}=$ (A) -25 (B) -3.57 (C) 0.20 (D) 0.82 (E) 1.40

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Problem 397

$a b c^{2}=0$ , evidents relaţia de deman şirurile: $\left(c_{n}\right)_{n \geq 1,}$ geometrice. $b_{n}$ ) are raţia ade de $n$ . Avem? (1) Progresia geometrică $\left(b_{n}\right)_{n \geq 1}$ de raţie $q$ este definită prin anumite elemente date. Determinaţi in fiećcare din cazuri, elementele cerute. a) $q=4, n=8, b_{8}=49152$ . Calculați $b_{1}$ şi $S_{8}$ ; b) $q=2, n=9, S_{9}=1533$ . Calculați $b_{1}$ şi $b_{9}$ ; c) $b_{1}=1, b_{n}=-512, S_{n}=-341$ . Calculați $n$ ; d) $b_{6}-b_{4}=216, b_{3}-b_{1}=8, S_{n}=40$ . Calculatii $b_{1}, q, n$ ; e) $S_{2}=3, S_{3}=7$ . Calculaţi $S_{4}$ ; f) $b_{4}=8, \frac{S_{2}}{S_{4}}=\frac{1}{5}$ . Calculati $b_{1}, q$ ; g) $b_{10}=3 \sqrt{3} b_{7}, S_{3}=4+\sqrt{3}$ . Calculaţi $S_{9}$ ; h) $S_{10}=33 \cdot S_{5}$ . Calculaţi $\frac{S_{12}}{S_{6}}$ .

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Problem 398

Listen
What expression do we get when we made a pattern by adding consecutive odd whole numbers? None of these possibilities are correct. $n+n$ $2 n$ $n^{2}$ $1+3+5+7+\ldots$

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Problem 399

An iterative formula is shown below. $x_{n+1}=\sqrt{8 x_{n}-5}$
Starting with $x_{1}=3$ , calculate the values of $x_{2}, x_{3}$ and $x_{4}$ . Give your answers to 3 d.p.

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Problem 400

Use a Maclaurin series in this table to obtain the Maclaurin series for the given function. $\begin{array}{r} f(x)=x^{2} \ln \left(1+x^{3}\right) \\ \sum_{n=1}^{\infty}(\square) \end{array}$

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Sequences & Series

Problem 301

Find a formula for the nthn^{\text{th}}nth term ana_{n}an​ of the sequence 3,−3,−9,…3, -3, -9, \ldots3,−3,−9,…

Problem 302

Find a formula for the nthn^{\text{th}}nth term ana_{n}an​ of the sequence: 23, 27, 31, ...

Problem 303

Find the sums of these sequences: a. 1+2+3+…+10021+2+3+\ldots+10021+2+3+…+1002 b. 1+3+5+…+9991+3+5+\ldots+9991+3+5+…+999 c. 6+13+20+…+7066+13+20+\ldots+7066+13+20+…+706 d. 494+489+484+…+4494+489+484+\ldots+4494+489+484+…+4

Problem 304

Find the sums of these sequences: a. 1+2+3+…+9991+2+3+\ldots+9991+2+3+…+999 b. 1+3+5+…+10031+3+5+\ldots+10031+3+5+…+1003 c. 4+9+14+…+4894+9+14+\ldots+4894+9+14+…+489 d. 293+290+287+…+2293+290+287+\ldots+2293+290+287+…+2 a. The sum is:

Problem 305

Find the sums of these sequences using Gauss's method: a. 1+2+3+…+10021+2+3+\ldots+10021+2+3+…+1002 b. 1+3+5+…+9991+3+5+\ldots+9991+3+5+…+999 c. 6+13+20+…+7066+13+20+\ldots+7066+13+20+…+706 d. 494+489+484+…+4494+489+484+\ldots+4494+489+484+…+4

Problem 306

Problem 307

Find the 100th term of the sequence 7,74,77,710,…7, 7^{4}, 7^{7}, 7^{10}, \ldots7,74,77,710,… in exponential notation.

Problem 308

Find the 100th term of the sequence 7,74,77,710,…7, 7^{4}, 7^{7}, 7^{10}, \ldots7,74,77,710,… and express it in exponential form.

Problem 309

Find the nnnth term of the sequence: 198+(n+4)⋅231198 + (n+4) \cdot 2^{31}198+(n+4)⋅231.

Problem 310

Find the nth term of the sequence 7,74,77,710,…7, 7^{4}, 7^{7}, 7^{10}, \ldots7,74,77,710,… and simplify your answer.

Problem 311

Find the relationship between consecutive numbers in the sequence: 2,6,18,54,162,…2, 6, 18, 54, 162, \ldots2,6,18,54,162,…

Problem 312

Find the relationship between consecutive numbers in the sequence: 4,12,36,108,324,…4, 12, 36, 108, 324, \ldots4,12,36,108,324,…

Problem 313

Shannon has job offers: Company ABC: \$45,500 with 8\% raise yearly; Company XYZ: \$62,000 with 3\% raise. After 15 years, which is better?

Problem 314

Problem 315

Problem 316

A cab charges \4permile,withanarithmeticfare.Findfaresfor4,5,and6miles:4 per mile, with an arithmetic fare. Find fares for 4, 5, and 6 miles: 4permile,withanarithmeticfare.Findfaresfor4,5,and6miles:\mathrm{f}(4),, ,\mathrm{f}(5),, ,\mathrm{f}(6)$. Write a recursive rule.

Problem 317

What comes next in the series: a3Z,b6Y,c9X,d12W,e15V,f18Ua 3 Z, b 6 Y, c 9 X, d 12 W, e 15 V, f 18 Ua3Z,b6Y,c9X,d12W,e15V,f18U? Options: A. g21T B. 924U924 \mathrm{U}924U C. G24 T\mathrm{G} 24 \mathrm{~T}G24 T D. g21tg 21 tg21t E. f21 T\mathrm{f} 21 \mathrm{~T}f21 T F. g24tg 24 tg24t

Problem 318

Find the sixth term in the sequence starting with 5, where each term is 2×previous−42 \times \text{previous} - 42×previous−4.

Problem 319

Write the sum using sigma notation: 6−8+10−12+14−16+18=∑n=1A6-8+10-12+14-16+18=\sum_{n=1}^{A}6−8+10−12+14−16+18=n=1∑A​ , where A=B=\begin{array}{l} A= \\ B= \end{array}A=B=​

Problem 320

Find the 82nd term of the arithmetic sequence −10,6,22,…-10,6,22, \ldots−10,6,22,…

Problem 321

How many terms are in this series? * ∑n=2918(0.1)n\sum_{n=2}^{9} 18(0.1)^{n}n=2∑9​18(0.1)n

Problem 322

Problem 323

Problem 324

Time left 0:06:46 HideBy using proof by induction specify the firbe arrec sieps iv prove that P(n)P(n)P(n) is true such that P(n)P(n)P(n) is: 8+16+24+⋯+8n=8n(n+1)28+16+24+\cdots+8 n=\frac{8 n(n+1)}{2}8+16+24+⋯+8n=28n(n+1)​ is true for all positive integers (n≥1)(n \geq 1)(n≥1)

Problem 325

Identify the incorrect expression for the sum −x∑r=03xr+1r!-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}−xr=0∑3​r!xr+1​ from the options given.

Problem 326

Write an equation for the sequence 7,9,11,13,15,…7, 9, 11, 13, 15, \ldots7,9,11,13,15,… and find the 20th term.

Problem 327

Find an equation for the sequence 7,9,11,13,15,…7, 9, 11, 13, 15, \ldots7,9,11,13,15,… and determine the 20th term.

Problem 328

Find the explicit formula for the sequence: 15, 19, 23, 27, ...

Problem 329

Find the recursive formula for the sequence: 15, 19, 23, 27, ...

Problem 330

Find the recursive formula for the sequence where a1=−1a_{1}=-1a1​=−1 and d=−10d=-10d=−10.

Problem 331

Find the explicit formulas for the sequences with: 9. a1=−1a_1 = -1a1​=−1, d=−10d = -10d=−10 10. a1=−3a_1 = -3a1​=−3, d=−9d = -9d=−9

Problem 332

Find the explicit formula for the sequence with a1=−1a_{1}=-1a1​=−1 and common difference d=−10d=-10d=−10.

Problem 333

Problem 334

Problem 335

Problem 336

Problem 337

Problem 338

The nth n^{\text {th }}nth term rule for a sequence is T(n)=n2+4T(n)=n^{2}+4T(n)=n2+4 Work out the 5th 5^{\text {th }}5th term of this sequence.

Problem 339

The nth n^{\text {th }}nth term rule for a sequence is T(n)=n2+3T(n)=n^{2}+3T(n)=n2+3Work out the 5th 5^{\text {th }}5th term of this sequence.

Problem 340

Problem 341

8. Write the explicit and recursive formula for the sequence −4,−1,2,…-4,-1,2, \ldots−4,−1,2,…

Problem 342

Write out the first three terms and the last term. Then use the formula for the sum of the first nnn terms of an arithmetic sequence to find the indicated sum. ∑i=125(9i−2)\sum_{i=1}^{25}(9 i-2)i=1∑25​(9i−2)

Problem 343

Problem 344

Consider the following series. ∑n=1∞3n2−14n5+3n+1\sum_{n=1}^{\infty} \frac{3 n^{2}-1}{4 n^{5}+3 n+1}n=1∑∞​4n5+3n+13n2−1​Use the Limit Comparison Test to complete the limit.

Problem 345

Determine whether the following series converges or diverges. ∑n=1∞(−1)nsin⁡(1n)\sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{1}{n}\right)n=1∑∞​(−1)nsin(n1​)Input CCC for convergence and DDD for divergence: □\square□Note: You have only one chance to enter your answer.

Problem 346

Find a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence $3, -3, -9, \ldots$

Find a formula for the $n^{\text{th}}$ term $a_{n}$ of the sequence: 23, 27, 31, ...

Find the sums of these sequences: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Find the sums of these sequences: a. $1+2+3+\ldots+999$ b. $1+3+5+\ldots+1003$ c. $4+9+14+\ldots+489$ d. $293+290+287+\ldots+2$ a. The sum is:

Find the sums of these sequences using Gauss's method: a. $1+2+3+\ldots+1002$ b. $1+3+5+\ldots+999$ c. $6+13+20+\ldots+706$ d. $494+489+484+\ldots+4$

Find the 100th term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ in exponential notation.

Find the 100th term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ and express it in exponential form.

Find the $n$ th term of the sequence: $198 + (n+4) \cdot 2^{31}$ .

Find the nth term of the sequence $7, 7^{4}, 7^{7}, 7^{10}, \ldots$ and simplify your answer.

Find the relationship between consecutive numbers in the sequence: $2, 6, 18, 54, 162, \ldots$

Find the relationship between consecutive numbers in the sequence: $4, 12, 36, 108, 324, \ldots$

A cab charges \ $4 per mile, with an arithmetic fare. Find fares for 4, 5, and 6 miles:$ \mathrm{f}(4) $,$ \mathrm{f}(5) $,$ \mathrm{f}(6)$. Write a recursive rule.

What comes next in the series: $a 3 Z, b 6 Y, c 9 X, d 12 W, e 15 V, f 18 U$ ? Options: A. g21T B. $924 \mathrm{U}$ C. $\mathrm{G} 24 \mathrm{~T}$ D. $g 21 t$ E. $\mathrm{f} 21 \mathrm{~T}$ F. $g 24 t$

Find the sixth term in the sequence starting with 5, where each term is $2 \times \text{previous} - 4$ .

Write the sum using sigma notation: $6-8+10-12+14-16+18=\sum_{n=1}^{A}$ , where $\begin{array}{l} A= \\ B= \end{array}$

Find the 82nd term of the arithmetic sequence $-10,6,22, \ldots$

How many terms are in this series? * $\sum_{n=2}^{9} 18(0.1)^{n}$

Time left 0:06:46 Hide
By using proof by induction specify the firbe arrec sieps iv prove that $P(n)$ is true such that $P(n)$ is: $8+16+24+\cdots+8 n=\frac{8 n(n+1)}{2}$ is true for all positive integers $(n \geq 1)$

Identify the incorrect expression for the sum $-x \sum_{r=0}^{3} \frac{x^{r+1}}{r !}$ from the options given.

Write an equation for the sequence $7, 9, 11, 13, 15, \ldots$ and find the 20th term.

Find an equation for the sequence $7, 9, 11, 13, 15, \ldots$ and determine the 20th term.

Find the recursive formula for the sequence where $a_{1}=-1$ and $d=-10$ .

Find the explicit formulas for the sequences with:
9. $a_1 = -1$ , $d = -10$
10. $a_1 = -3$ , $d = -9$

Find the explicit formula for the sequence with $a_{1}=-1$ and common difference $d=-10$ .

The $n^{\text {th }}$ term rule for a sequence is $T(n)=n^{2}+4$ Work out the $5^{\text {th }}$ term of this sequence.

The $n^{\text {th }}$ term rule for a sequence is $T(n)=n^{2}+3$
Work out the $5^{\text {th }}$ term of this sequence.

8. Write the explicit and recursive formula for the sequence $-4,-1,2, \ldots$

Write out the first three terms and the last term. Then use the formula for the sum of the first $n$ terms of an arithmetic sequence to find the indicated sum. $\sum_{i=1}^{25}(9 i-2)$

Consider the following series. $\sum_{n=1}^{\infty} \frac{3 n^{2}-1}{4 n^{5}+3 n+1}$
Use the Limit Comparison Test to complete the limit.

Determine whether the following series converges or diverges. $\sum_{n=1}^{\infty}(-1)^{n} \sin \left(\frac{1}{n}\right)$
Input $C$ for convergence and $D$ for divergence: $\square$
Note: You have only one chance to enter your answer.

(1 point)
Classify each series as absolutely Convergent, Conditionally Convergent, or Divergent. (a) $\sum_{k=1}^{\infty} \frac{(-7)^{k}}{k^{2}}$ ? (b) $\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{k!}$ ?

Étudiez la convergence (absolue ou conditionnelle) de la série alternée : $\sum_{n=1}^{\infty}(-1)^{n}\left(\frac{n}{7 n^{2}+3}\right)$

Find the formula for the $n^{\text{th}}$ term of the sequence $5, 9, 13, 17, 21, \ldots$ . What is $a_{n}=$ ?

Find the 10th term in the arithmetic sequence $11, 19, 27, 35, 43, \ldots$ . Options: $91, 75, 100, 83$ .

Find the sum of the series: $\sum_{n=0}^{\infty} \frac{n^{10}}{10^{n}}$ .

Find the interval of convergence for the series $\sum_{n=1}^{\infty} \frac{(x-1)^{n}}{3^{n}}$ .

Marika starts sprinting 100 yards and adds 4 yards daily for 21 days. Find her distance on day 21 using $a_{n}=100+(n-1)4$ . Options: A. 200 B. 184 C. 221 D. 180.

Find the common ratio of the geometric sequence where $\mathrm{a}_{1}=1$ and $\mathrm{a}_{9}=25$ .

수렴하는 수열을 고르세요: ᄀ. $\left\{\frac{1}{2 n+1}\right\}$ , ㄴ. $\left\{\frac{(-1)^{n}}{n}\right\}$ , ᄃ. $\left\{(-1)^{n}+2\right\}$ .

Find the pattern in the sequence $-2, -8, -14, -20, \ldots$ .

$\sum_{k=1}^{n}\left(b_{k}-9 a_{k}\right)$ where $a_k = 7$ and $b_k = 30$ .

Write the sum using sigma notation: $7+9+11+13+\ldots+23=\sum_{n=1}^{A} B$ , where $A=$ $\square$ $B=$ $\square$

Find the sum of the series $S=\sum_{i=1}^{\infty} x^{4 i},|x|<1$ $S=$

5. What are the first three terms of the Arithmetic Sequence $a_{n}=11-3(n-1) ?$ $0,8,16$ $11,8,5$ $1,2,3$ $8,5,2$

Find the sum of the given finite geometric series. $6+\frac{6}{3}+\frac{6}{9}+\frac{6}{27}+\ldots+\frac{6}{59049}$
The sum of the finite geometric series is $\square$ (Type an integer or a simplified fraction.)

Find the explicit formula for the sequence $-7, -4, -7, 2, 5$ . Choose from A, B, C, or D.

Find the 500th term of the sequence $24, 31, 38, 45, 52, \ldots$ using $a_{n}=a_{1}+(n-1) \cdot d$ . A. 3493 B. 3545 C. 3517 D. 3524

Find the formula for Marika's daily sprint distance after 21 days: A. $a_{n}=100+(n-1) 4$ B. $a_{n}=4+(n-1) 100$ C. $a_{n}=100+(n-1) 21$ D. $a_{n}=21+(n-1) 4$

Zra sprints 100 yards and adds 5 yards daily for 21 days. Find his distance on day 21 using $a_{n}=100+(n-1)5$ . Options: A. 200 B. 100 C. 225 D. 205