Question 145
Let $a_1, a_2, a_3, \dots$ be an arithmetic sequence. For an arbitrary positive integer $k$ denote $S_k = a_1 + a_2 + \dots + a_k$. Prove that
$$
S_n - S_m = \frac{n-m}{n+m} S_{n+m}
$$
holds for all positive integers $m$ and $n$.
$$
S_n - S_m = \frac{n-m}{n+m} S_{n+m}
$$
holds for all positive integers $m$ and $n$.