Problem:
For a finite sequence A=(a1,a2,…,an) of numbers, the Cesàro sum of A is defined to be
nS1+S2+⋯+Sn
where Sk=a1+a2+⋯+ak(1≤k≤n). If the Cesàro sum of the 99-term sequence (a1,a2,…,a99) is 1000, what is the Cesàro sum of the 100-term sequence (1,a1,a2,…,a99)?
Answer Choices:
A. 991
B. 999
C. 1000
D. 1001
E. 1009
Solution:
Since 99S1+S2+⋯+S99=1000,S1+S2+⋯+S99=99000. Thus
1001+(1+S1)+(1+S2)+⋯+(1+S99)=100100+(S1+S2+⋯+S99)=100100+99000=991
OR
Let C be the Cesàro sum of the n-term sequence (b1,b2,…,bn). Then
b1+(b1+b2)+(b1+b2+b3)+⋯+(b1+b2+⋯+bn)=nC.
The Cesàro sum of the (n+1)-term sequence k,b1,b2,…,bn is then
n+1k+(k+b1)+(k+b1+b2)+⋯+(k+b1+b2+⋯+bn)=n+1(n+1)k+(b1+(b1+b2)+⋯+(b1+b2+⋯+bn))=n+1(n+1)k+nC=k+n+1nC.
In the problem, we have n=99,C=1000, and k=1, which gives a Cesàro sum of 991.
Note. Cesàro sums are named after the nineteenth century mathematician E. Cesàro. Cesàro sums arise in many areas of mathematics such as in the study of Fourier Series.