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Front to back (Posted on 2007-03-18) Difficulty: 3 of 5
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.

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Solution induction | Comment 2 of 3 |
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) + n+4n+4
=sum2(n) + (n+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
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