Problem 48 - Beginning and Intermediate Algebra

Question

Find the sum of the first nine terms of the arithmetic sequence with first term 2 and last term 34

Step-by-Step Solution

Answer

The sum of the first nine terms of the arithmetic sequence is 162.

Step 1: Analyze given information

We are given that the first term, denoted as \(a_1\), is 2 and the last term, denoted as \(a_9\), is 34. We're supposed to find the sum of these 9 terms.

Step 2: Find the common difference (d)

Since this sequence is arithmetic, there is a common difference (denoted as d) between consecutive terms. As there are 8 intervals between the first and the ninth term, we can calculate the common difference (d) with the following formula: \[ d = \frac{a_9 - a_1}{8} \] Plug in the known values: \[ d = \frac{34 - 2}{8} = \frac{32}{8} = 4 \] Thus, the common difference is 4.

Step 3: Find the sum of the first nine terms

Now that we know the common difference, we can find the sum of the first 9 terms using the formula for the arithmetic sequence sum: \[ S_n = \frac{n(a_1 + a_n)}{2} \] Here, \(n\) represents the number of terms, and we have all the values needed to solve for the sum: \[ S_9 = \frac{9(2 + 34)}{2} \]

Step 4: Calculate the sum

Plug in the values and solve for the sum: \[ S_9 = \frac{9(36)}{2} = \frac{324}{2} = 162 \] Thus, the sum of the first nine terms of this arithmetic sequence is 162.