For a finite sequence A=(a_1,a_2,…,a_n) of numbers, the Cesàro sum of A is defined to be (S_1+S_2+\cdots+S_n)/(n), where \beginaligned S_k&=a_1+a_2+a_3+\cdots+a_k\ &\qquad(1≤ k≤ n). \endaligned If the Cesàro sum of the 99 -term sequence (a_1,a_2,…,a_99) is 1000, what is the Cesàro sum of the 100 -term sequence (1,a_1,a_2,…,a_99)?

1992 AMC 12 Problem 21 — Solution

1992 AMC 12 Problem 21

Attempt Problem 21 of the 1992 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1992 AMC 12 solutions, or check the answer key.

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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 $(1,a_1,a_2,\ldots,a_{99})?$

$991$

$999$

$1000$

$1001$

$1009$

Answer: A

Concepts:partial sums sequence transformation weighted average

Difficulty rating: 2000

Small Hint:

Convert the first Cesàro sum into the value of $S_1+\cdots+S_{99}$

Big Hint:

Each new partial sum is $1$ plus an old partial sum, with one initial partial sum equal to $1$

Solution:

The given condition says $S_1+\cdots+S_{99}=99{,}000.$ For the new sequence, the partial sums are $1,1+S_1,\ldots,1+S_{99}.$ Their sum is $100+(S_1+\cdots+S_{99})=99{,}100.$ Dividing by $100$ gives the new Cesàro sum $991.$

Thus the correct answer is A.

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