Proof with mathematic induction

[katy_042]

Hello guys. I need help to solve this formula, probably using mathematic induction but I don't really know how. Thanks for your help

 

[lev888]

Hello guys. I need help to solve this formula, probably using mathematic induction but I don't really know how. Thanks for your help

Ok, please post the first 2 steps of the proof. Then we'll work on the 3rd.

 

[HallsofIvy]

"Proof by induction" involves
1) Prove it is true when n= 1. That's easy, the left side is just 1 and the right side is $\displaystyle \left(\frac{1}{2}(1)(1+1)\right)^2= ((1/2)(1)(2))^2= 1^2= 1$.

2) Prove that if this is true for n= k then it is also true for n= k+1.
For n= k that is $\displaystyle \sum_{i=1}^{k} i^3= \left(\frac{1}{2}k(k+1)\right)^2$
and you want to use that to show that
$\displaystyle \sum_{i=1}^{k+1} i^3= \left(\frac{1}{2}k(k+1)\right)^2$.

The reason so many "induction problems" involve sums is because
the sum to k+ 1 is just the sum to k plus one more term. Here that is
$\displaystyle \sum_{i=1}^{k} i^3+ (k+1)^3= \left(\frac{1}{2}k(k+1)\right)^2+ (k+1)^3$.

 

[lex]

and you want to use that to show that
[MATH] \sum_{i=1}^{k+1} i^3= \left(\frac{1}{2}k(k+1)\right)^2.[/MATH]

@HallsofIvy
Did you mean:
[MATH] \sum_{i=1}^{k+1} i^3= \left(\frac{1}{2}(k+1)(k+2)\right)^2[/MATH]?