Question $2+5+8+\cdots+x=155$

Studdy Solution

STEP 1

What is this asking? Find the value of $x$ that makes the sum of this arithmetic sequence equal to 155. Watch out! Don't forget that $x$ has to be a term in the sequence, not just any number that makes the sum work!

STEP 2

1. Define the sequence
2. Find the number of terms
3. Solve for $x$

STEP 3

Alright, let's break this problem down!
We've got an arithmetic sequence here.
Remember, an arithmetic sequence is just a list of numbers where the difference between any two consecutive numbers is the same.
That difference is called the common difference.

STEP 4

In our sequence, $2, 5, 8, \dots, x$ , we can see that the common difference is $5 - 2 = 3$ .
So, we're adding 3 each time!

STEP 5

The general formula for an arithmetic sequence is $a_n = a_1 + (n-1)d$ , where $a_n$ is the $n$ th term, $a_1$ is the first term, $n$ is the number of terms, and $d$ is the common difference.

STEP 6

In our case, $a_1 = 2$ and $d = 3$ .
So, our formula becomes $a_n = 2 + (n-1) \cdot 3$ .

STEP 7

We know that the sum of an arithmetic series is given by $S_n = \frac{n}{2}(a_1 + a_n)$ , where $S_n$ is the sum of the first $n$ terms.
We're given that $S_n = 155$ .
We also know that $a_1 = 2$ and $a_n = x$ .

STEP 8

Substituting these values into the sum formula, we get $155 = \frac{n}{2}(2 + x)$ .

STEP 9

Now, remember our formula for the $n$ th term from before: $a_n = 2 + (n-1) \cdot 3$ .
Since $a_n = x$ , we can write $x = 2 + (n-1) \cdot 3$ .

STEP 10

Let's substitute this expression for $x$ into our sum equation: $155 = \frac{n}{2}(2 + 2 + (n-1) \cdot 3)$ .

STEP 11

Simplifying, we get $155 = \frac{n}{2}(4 + 3n - 3)$ , which further simplifies to $155 = \frac{n}{2}(3n + 1)$ .

STEP 12

Multiplying both sides by 2 gives us $310 = n(3n+1)$ , or $3n^2 + n - 310 = 0$ .

STEP 13

Now, we can solve this quadratic equation for $n$ .
Factoring, we find $(3n+31)(n-10)=0$ .
So, $n=10$ or $n = -\frac{31}{3}$ .
Since the number of terms must be a positive integer, we have $n = \textbf{10}$ !

STEP 14

We found that $n=10$ , which means $x$ is the 10th term in the sequence.
We can use our formula for the $n$ th term to find $x$ : $x = a_{10} = 2 + (10-1) \cdot 3$ .

STEP 15

Calculating this gives us $x = 2 + 9 \cdot 3 = 2 + 27 = \textbf{29}$ .

STEP 16

The value of $x$ is 29.

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