Problem: Find the sum
1(3)1+3(5)1+…+(2n−1)(2n+1)1+…+255(257)1
Answer Choices:
A. 255127
B. 255128
C. 21
D. 257128
E. 257129
Solution:
Since n(n+2)1=21(n1−n+21), the desired sum is
21(1−31+31−51+…+2N−11−2N+11)=21(1−2N+11)=2N+1N,
with N=128.
OR
Since n(n−k)1+n(n+k)1=(n−k)(n+k)2, the desired sum is
2(1(5)1+5(9)1+…+253(257)1)4(1(9)1+9(17)1+…+249(257)1)==128(1(257)1).
OR
The result obtained in the first solution may also be obtained by mathematical induction.