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Chapter 15: Problem 48

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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: \[ S_9 = \frac{9(2 + 34)}{2} \]
04

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.

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Most popular questions from this chapter

Use the formula for \(S_{n}\) to find the sum of the terms of each geometric sequence. $$\sum_{i=1}^{4} 10\left(-\frac{2}{5}\right)^{i}$$

Find the sum of the first four terms of the geometric sequence with \(a_{1}=6\) and \(r=3\)

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