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Prove the following using mathematical induction

0 votes

\sum_{i=1}^{n}i^3 = (\sum_{i=1}^{n}i)^2

asked Oct 15, 2015 in Number Theory by chstw (1,320 points)

1 Answer

0 votes

For n=1, it is obvious.

Assuming that

\sum_{i=1}^{n}i^3 = (\sum_{i=1}^{n}i)^2

We want to prove that

\sum_{i=1}^{n+1}i^3 = (\sum_{i=1}^{n+1}i)^2

Left side = 

\sum_{i=1}^{n}i^3 + (n+1)^3 = (\sum_{i=1}^{n}i)^2 + (n+1)^3 \text { by our assumption}

Right side = 

(\sum_{i=1}^{n}i + (n+1))^2 = (\sum_{i=1}^{n}i )^2 + 2 (\sum_{i=1}^{n}i )(n+1) + (n+1)^2

All we need to do is to prove 

(n+1)^3 = 2 (\sum_{i=1}^{n}i )(n+1) + (n+1)^2

That is 

(n+1)^2 = 2 (\sum_{i=1}^{n}i ) + (n+1) 

which is true since

2 (\sum_{i=1}^{n}i ) = n(n+1)

The proof is done.

answered Oct 15, 2015 by w1002 (660 points)
edited Oct 15, 2015 by w1002
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