등차수열 $\left\{a_{n}\right\}$ 에 대하여
\[
a_{14}-a_{8}=24, a_{8}+a_{2}=10
\]
일 때, 수열 $\left\{a_{n}\right\}$ 의 첫째항부터 제 15 항까지의 합은?
(1) 195
(2) 210
(3) 225
(4) 240
(5) 255

Step 1 ：Given an arithmetic sequence \(\left\{a_{n}\right\}\) with the conditions: \(a_{14} - a_{8} = 24\) and \(a_{8} + a_{2} = 10\).

Step 2 ：Express the terms in the sequence using the common difference \(d\) and the first term \(a_1\): \(a_2 = a_1 + d\), \(a_8 = a_1 + 7d\), and \(a_{14} = a_1 + 13d\).

Step 3 ：Rewrite the given conditions using the expressions for the terms: \((a_1 + 13d) - (a_1 + 7d) = 24\) and \((a_1 + 7d) + (a_1 + d) = 10\).

Step 4 ：Solve the system of equations to find \(a_1\) and \(d\): \(a_1 = -11\) and \(d = 4\).

Step 5 ：Use the formula for the sum of an arithmetic sequence to find the sum of the first 15 terms: \(S_n = \frac{n}{2}(2a_1 + (n-1)d)\), where \(n = 15\), \(a_1 = -11\), and \(d = 4\).

Step 6 ：Calculate the sum of the first 15 terms: \(S_n = 255\).

Step 7 ：\(\boxed{255}\) is the sum of the first 15 terms of the sequence.