# need induction proof help

• Nov 9th 2012, 05:26 AM
Petrus
need induction proof help
$\displaystyle \sum_{k=0}^nk^3=\left(\sum_{k=0}^nk \right)^2$
i did use the base case and find it true
1=1
0^3+1^3=0+1*0+1
im not good with latex but in my paper the right side just to make it more "simple" for me i did write it as 2 times that instend to have ^2...
ima proof for k+1 and need tips :S im lost
• Nov 9th 2012, 09:21 AM
fkf
Re: need induction proof help
You can use the fact that the right side is just a square of a ordinary arithmetic sum.
sum = n/2(a_1+a_n)
• Nov 9th 2012, 10:01 AM
Plato
Re: need induction proof help
Quote:

Originally Posted by Petrus
$\displaystyle \sum_{k=1}^nk^3=\left(\sum_{k=1}^nk \right)^2$

If I were you, I would note that $\displaystyle \left(\sum_{k=1}^nk \right)^2=\frac{n^2(n+1)^2}{4}$

Then the inductive step becomes
$\displaystyle \sum_{k=1}^{n+1}k^3=\sum_{k=1}^nk^3+(n+1)^3=\frac{ n^2(n+1)^2}{4}+(n+1)^3$
• Nov 9th 2012, 10:11 AM
MarkFL
Re: need induction proof help
This is an exercise I gave to the OP to practice induction, and the suggestion given by Plato is exactly what I had in mind.