Question 126
Given an arithmetic sequence $a_{1}, a_{2}, a_{3}, \ldots$ Let us denote by $s_{i}$ the sum of the first $i$ terms of this sequence, by $s_{j}$ the sum of the first $j$ terms and by $s_{k}$ the sum of the first $k$ terms. Prove that the value of the expression
$$
\frac{s_{i}}{i}(j-k)+\frac{s_{j}}{j}(k-i)+\frac{s_{k}}{k}(i-j)
$$
it does not depend on the choice of numbers $i, j$ and $k$, nor on the sequence.
$$
\frac{s_{i}}{i}(j-k)+\frac{s_{j}}{j}(k-i)+\frac{s_{k}}{k}(i-j)
$$
it does not depend on the choice of numbers $i, j$ and $k$, nor on the sequence.