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.