Suppose one sequence is formed by multiplying termwise the sequence 1,2,...n by the sequence n,n-1,...1 and another sequence is formed by multiplying termwise the sequence 2,4,...n by the sequence 2,4,...n.
Show that for even n, the sum of each sequence is the same.
Writing n=2K, the first sum is the sum for I=1 to K of I times (2K+1-I), or (2K+1) times the sum of all numbers from 1 to 2K, minus the sum of the squares of all numbers from 1 to 2K, which eventually works out to 2/3.K.(K+1).(2K+1).
The second sum is 4 times the sum of squares of the numbers from 1 to K, which comes out to the same value.