Question 21
For a finite sequence $A=(a_1,a_2,\ldots,a_n)$ of numbers, the *Cesàro sum* of $A$ is defined to be $$
\frac{S_1+S_2+\cdots+S_n}{n},
$$ where $$
\begin{aligned} S_k&=a_1+a_2+a_3+\cdots+a_k\\ &\qquad(1\le k\le n). \end{aligned}
$$ If the Cesàro sum of the $99$-term sequence $(a_1,a_2,\ldots,a_{99})$ is $1000,$ what is the Cesàro sum of the $100$-term sequence $(1,a_1,a_2,\ldots,a_{99})?$
\frac{S_1+S_2+\cdots+S_n}{n},
$$ where $$
\begin{aligned} S_k&=a_1+a_2+a_3+\cdots+a_k\\ &\qquad(1\le k\le n). \end{aligned}
$$ If the Cesàro sum of the $99$-term sequence $(a_1,a_2,\ldots,a_{99})$ is $1000,$ what is the Cesàro sum of the $100$-term sequence $(1,a_1,a_2,\ldots,a_{99})?$