A high school competition-level problem concerning sum and sequence

Question

Given the sum of first N elements of sequence $A$: $S_{n} = n^{2} + 3n + 4$.
Compute $A_{1} + A_{3} + \ldots + A_{21}$.

I know this problem can be tackled by carefully calculating each value of the sequence. But I wonder what are the better ways to solve it.

Thanks in advance!

Answer 1

$A_n=S_n-S_{n-1}=2n+2$. Now one has to find a sum of an arithmetic progression $4+8+\dots+44$.

Answer 2

Determining a nice form for A(n) is a good start. This is S(n) - S(n-1) = 2n + 2 . Now the desired sum is sum {k= 1 to 10} A(2k-1) = sum {k=1 to 10} 4k = 4 * 10 * 11 / 2 = 220 .