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.

It is easily tru for n=0 (both sums =0)

Now for n+2

Sum1(n+2) = sum1(n) + 2*sum(i=1..n)(i) + n+2 + 2(n+1)

(each term increases by two + two new terms)

=sum1(n) + n(n+1) + 3n + 4 = sum1(n) + n²+4n+4
=sum2(n) + (n+2)² = sum2(n+2)

so by induction it is proven for all even n >=0

(sum2 substitution by inductive hypothesis)

Posted by Joel on 2007-03-18 13:16:00