Question

question
evaluate the following sum.
17 + 22 + 27 +...+ 57

Explanation:

Step1: Identify the arithmetic - sequence

This is an arithmetic sequence with first term \(a_1 = 17\), common difference \(d=5\), and last term \(a_n = 57\).

Step2: Find the number of terms \(n\)

Use the formula \(a_n=a_1+(n - 1)d\). Substitute \(a_n = 57\), \(a_1 = 17\), and \(d = 5\) into it: \(57=17+(n - 1)\times5\).
First, simplify the equation: \(57-17=(n - 1)\times5\), so \(40=(n - 1)\times5\). Then \(n - 1=\frac{40}{5}=8\), and \(n=9\).

Step3: Calculate the sum \(S_n\)

Use the sum formula for an arithmetic sequence \(S_n=\frac{n(a_1 + a_n)}{2}\). Substitute \(n = 9\), \(a_1 = 17\), and \(a_n = 57\) into it: \(S_9=\frac{9\times(17 + 57)}{2}\).
First, calculate the sum inside the parentheses: \(17+57 = 74\). Then \(S_9=\frac{9\times74}{2}=9\times37 = 333\).

Answer:

333