Understanding Induction and Finding Formulas for Summations

[carlosbgois]

Homework Statement

1b) Prove by induction: $1^{3}+...+n^{3}=(1+...+n)^{2}$
2a) Find a formula for: $\sum^{n}_{i=1}(2i-1)$

Homework Equations

There's a Hint for 2a): 'What to this expression have to do with $1+2+3+...+2n$?'

The Attempt at a Solution

In 2a) I've got near the answer, when comparing with the given one, but I can't understand the last thing he does. The solution in the book is:

$\sum^{n}_{i=1}(2i-1)=1+2+3+...+2n-2(1+...+n) =(2n)(2n+1)/2-n(n+1)$

And I couldn't understand how to make the second member become the third one, which goes directly to the answer $n^{2}$


Thanks

 

[gb7nash]

It's a well known fact that for positive integer n:

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

Use this to obtain the answer.

 

[206PiruBlood]

We can get the odd integers by first starting with all integers and removing those which are even.

 

[carlosbgois]

Got it, thanks.