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FEB 14, 2005 05:18 PM
revonrat said:
threejane said:
Prove that sum(i^3, 1..n) = (n^2*(n+1)^2)/4
By induction:
For n=1: obvious.
Assume that sum(i^3, 1...k) = (k^2*(k+1)^2)/4
Then sum(i^3, 1...(k+1)) = (k^2*(k+1)^2)/4 + (k+1)^3
= (k^4 + 2k^3 + k^2)/4 + (4k^3 + 12k^2 + 12k + 4)/4
= (k^4 + 6k^3 + 13k^2 + 12k + 4)/4
Now look at (k+1)^2*(k+2)^2/4 = (k^2 + 2k + 1)(k^2 + 4k + 4)/4
= (k^4 + 6k^3 + 13k^2 + 12k + 4)/4
So sum(i^3, 1...(k+1)) is equal to (k+1)^2*(k+2)^2/4, which completes the proof by induction.
Awe ... That one was for MisterSatan. At least if you were going to spoil it, you could have had the decency to be female
That's okay; I ignored it because I asked for a problem to solve, not prove.
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Venice
SUICIDEGIRL
Oregon, USA
FEB 14, 2005 02:47 PM