Math / Algebra

QuestionFind the sum. $\frac{1}{5} + \frac{4}{5} + \frac{4^2}{5} + \frac{4^3}{5} + \dots + \frac{4^{n-1}}{5}$ Complete the sum of the sequence. $S_n = \Box(\Box^n - 1)$ (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

What is this asking? We need to find the sum of a geometric series and express the result in a specific form. Watch out! It's easy to mess up the formula for the sum of a geometric series if we're not careful!
Also, remember that the first term isn't always the same as the common ratio.

STEP 2

1. Identify the series
2. Apply the formula
3. Simplify the result

STEP 3

Alright, let's dive in!
We've got a geometric series here.
How do we know?
Each term is obtained by multiplying the previous term by the same number.
That number is our common ratio, often denoted by $r$ .
In our case, $r = 4$ .
See how each term is 4 times the previous one?

STEP 4

Our first term, often denoted by $a$ , is $\frac{1}{5}$ .
So, we have $a = \frac{1}{5}$ .

STEP 5

We also know that there are $n$ terms in this series.

STEP 6

The sum of a finite geometric series is given by the formula: $S_n = \frac{a(r^n - 1)}{r - 1}$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

STEP 7

Let's plug in our values: $a = \frac{1}{5}$ , $r = 4$ , and we have $n$ terms. $S_n = \frac{\frac{1}{5}(4^n - 1)}{4 - 1}$

STEP 8

Let's simplify the denominator: $4 - 1 = 3$ . $S_n = \frac{\frac{1}{5}(4^n - 1)}{3}$

STEP 9

Now, let's multiply the fraction by $\frac{1}{3}$ , which is the same as dividing by 3.
Remember, dividing by 3 is the same as multiplying by its reciprocal, $\frac{1}{3}$ . $S_n = \frac{1}{5} \cdot \frac{1}{3} (4^n - 1)$

STEP 10

Multiplying the fractions $\frac{1}{5}$ and $\frac{1}{3}$ gives us: $S_n = \frac{1}{15}(4^n - 1)$

STEP 11

The sum is $S_n = \frac{1}{15}(4^n - 1)$ .

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QuestionFind the sum. $\frac{1}{5} + \frac{4}{5} + \frac{4^2}{5} + \frac{4^3}{5} + \dots + \frac{4^{n-1}}{5}$ Complete the sum of the sequence. $S_n = \Box(\Box^n - 1)$ (Type an integer or a simplified fraction.)

Studdy Solution

STEP 1

What is this asking? We need to find the sum of a geometric series and express the result in a specific form. Watch out! It's easy to mess up the formula for the sum of a geometric series if we're not careful!
Also, remember that the first term isn't always the same as the common ratio.

STEP 2

1. Identify the series
2. Apply the formula
3. Simplify the result

STEP 3

Alright, let's dive in!
We've got a geometric series here.
How do we know?
Each term is obtained by multiplying the previous term by the same number.
That number is our common ratio, often denoted by $r$ .
In our case, $r = 4$ .
See how each term is 4 times the previous one?

STEP 4

Our first term, often denoted by $a$ , is $\frac{1}{5}$ .
So, we have $a = \frac{1}{5}$ .

STEP 5

We also know that there are $n$ terms in this series.

STEP 6

The sum of a finite geometric series is given by the formula: $S_n = \frac{a(r^n - 1)}{r - 1}$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

STEP 7

Let's plug in our values: $a = \frac{1}{5}$ , $r = 4$ , and we have $n$ terms. $S_n = \frac{\frac{1}{5}(4^n - 1)}{4 - 1}$

STEP 8

Let's simplify the denominator: $4 - 1 = 3$ . $S_n = \frac{\frac{1}{5}(4^n - 1)}{3}$

STEP 9

Now, let's multiply the fraction by $\frac{1}{3}$ , which is the same as dividing by 3.
Remember, dividing by 3 is the same as multiplying by its reciprocal, $\frac{1}{3}$ . $S_n = \frac{1}{5} \cdot \frac{1}{3} (4^n - 1)$

STEP 10

Multiplying the fractions $\frac{1}{5}$ and $\frac{1}{3}$ gives us: $S_n = \frac{1}{15}(4^n - 1)$

STEP 11

The sum is $S_n = \frac{1}{15}(4^n - 1)$ .

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Studdy Solution

STEP 1

What is this asking? We need to find the sum of a geometric series and express the result in a specific form. Watch out! It's easy to mess up the formula for the sum of a geometric series if we're not careful!Also, remember that the first term isn't always the same as the common ratio.

STEP 2

1. Identify the series2. Apply the formula3. Simplify the result

STEP 3

Alright, let's **dive in**!We've got a geometric series here.How do we know?Each term is obtained by multiplying the previous term by the same number.That number is our **common ratio**, often denoted by rrr.In our case, r=4r = 4r=4.See how each term is 4 times the previous one?

STEP 4

Our **first term**, often denoted by aaa, is 15\frac{1}{5}51​.So, we have a=15a = \frac{1}{5}a=51​.

STEP 5

We also know that there are nnn terms in this series.

STEP 6

The **sum of a finite geometric series** is given by the formula: Sn=a(rn−1)r−1S_n = \frac{a(r^n - 1)}{r - 1}Sn​=r−1a(rn−1)​ where SnS_nSn​ is the sum of the first nnn terms, aaa is the first term, rrr is the common ratio, and nnn is the number of terms.

STEP 7

Let's **plug in** our values: a=15a = \frac{1}{5}a=51​, r=4r = 4r=4, and we have nnn terms. Sn=15(4n−1)4−1S_n = \frac{\frac{1}{5}(4^n - 1)}{4 - 1}Sn​=4−151​(4n−1)​

STEP 8

Let's **simplify** the denominator: 4−1=34 - 1 = 34−1=3. Sn=15(4n−1)3S_n = \frac{\frac{1}{5}(4^n - 1)}{3}Sn​=351​(4n−1)​

STEP 9

Now, let's **multiply** the fraction by 13\frac{1}{3}31​, which is the same as dividing by 3.Remember, dividing by 3 is the same as multiplying by its reciprocal, 13\frac{1}{3}31​. Sn=15⋅13(4n−1)S_n = \frac{1}{5} \cdot \frac{1}{3} (4^n - 1)Sn​=51​⋅31​(4n−1)

STEP 10

**Multiplying** the fractions 15\frac{1}{5}51​ and 13\frac{1}{3}31​ gives us: Sn=115(4n−1)S_n = \frac{1}{15}(4^n - 1)Sn​=151​(4n−1)

STEP 11

The sum is Sn=115(4n−1)S_n = \frac{1}{15}(4^n - 1)Sn​=151​(4n−1).

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

What is this asking? We need to find the sum of a geometric series and express the result in a specific form. Watch out! It's easy to mess up the formula for the sum of a geometric series if we're not careful!
Also, remember that the first term isn't always the same as the common ratio.

1. Identify the series
2. Apply the formula
3. Simplify the result

Alright, let's dive in!
We've got a geometric series here.
How do we know?
Each term is obtained by multiplying the previous term by the same number.
That number is our common ratio, often denoted by $r$ .
In our case, $r = 4$ .
See how each term is 4 times the previous one?

Our first term, often denoted by $a$ , is $\frac{1}{5}$ .
So, we have $a = \frac{1}{5}$ .

We also know that there are $n$ terms in this series.

The sum of a finite geometric series is given by the formula: $S_n = \frac{a(r^n - 1)}{r - 1}$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

Let's plug in our values: $a = \frac{1}{5}$ , $r = 4$ , and we have $n$ terms. $S_n = \frac{\frac{1}{5}(4^n - 1)}{4 - 1}$

Let's simplify the denominator: $4 - 1 = 3$ . $S_n = \frac{\frac{1}{5}(4^n - 1)}{3}$

Now, let's multiply the fraction by $\frac{1}{3}$ , which is the same as dividing by 3.
Remember, dividing by 3 is the same as multiplying by its reciprocal, $\frac{1}{3}$ . $S_n = \frac{1}{5} \cdot \frac{1}{3} (4^n - 1)$

Multiplying the fractions $\frac{1}{5}$ and $\frac{1}{3}$ gives us: $S_n = \frac{1}{15}(4^n - 1)$

The sum is $S_n = \frac{1}{15}(4^n - 1)$ .