# Mathematical Induction Question

• June 24th 2008, 04:19 AM
Mathematical Induction Question
Hello, I have this problem that I can't figure out. I don't think I have too much issue with a simple version of maths induction but this particular question I can't seem to figure out.

The question is:

Use mathematical induction to prove that for every positive integer
n

1^
3 + 2^3 + 3^3 + . . . + n^3 = ((n (n + 1))/2)^2

I can only get to this stage:

((n(n + 1))/2)^2 + (n + 1)^3 = (((n + 1)((n + 1) + 1))/2)^2

I would greatly appreciate it if some wiz could help me out.

Thanks.
• June 24th 2008, 04:30 AM
topsquark
Quote:

Hello, I have this problem that I can't figure out. I don't think I have too much issue with a simple version of maths induction but this particular question I can't seem to figure out.

The question is:

Use mathematical induction to prove that for every positive integer
n
1^
3 + 2^3 + 3^3 + . . . + n^3 = ((n (n + 1))/2)^2

I can only get to this stage:

((n(n + 1))/2)^2 + (n + 1)^3 = (((n + 1)((n + 1) + 1))/2)^2

I would greatly appreciate it if some wiz could help me out.

Thanks.

Expand the left hand side:
$\frac{1}{4} ( n^4 + 2n^3 + n^2 ) + (n^3 + 3n^2 + 3n + 1)$

$= \frac{1}{4} ( n^4 + 6n^3 + 13n^2 + 12n + 4 )$

Now the right hand side:
$\frac{1}{4}(n^2 + 2n + 1)(n^2 + 4n + 4) = \frac{1}{4} \left [ (n^4 + 4n^3 + 4n^2) +( 2n^3 + 8n^2 + 8n) + (n^2 + 4n + 4) \right ]$

$= \frac{1}{4} ( n^4 + 6n^3 + 13n^2 + 12n + 4 )$

-Dan
• June 24th 2008, 05:11 AM