double summation problem, index change formula

Question

I'm having difficulty with a problem from a textbook. The question asks to show the following is true:

https://i.stack.imgur.com/6dMNa.jpg

I am having difficulty going from the right hand side of the equation to the left hand side. Can someone help me out? Thanks,

(I don't have enough reputation points to post the image so hopefully the link works.)

Answer 1

important notice The indices $i$ are superfluous, let's drop them!
There's probably a typo : the $n$s should be $m$s

Equality works both ways. All you have to do to check this equality is to develop the left term.
Try with $m = 2$, then with $m = 3$...

You'll soon find a way to establish the recurrence :

If, for some $m$ we have $$\forall a_1, \dots, a_m, \ \big( \sum_{j = 1}^m a_j \big)^2 = \sum_{j = 1}^m a_j^2 + \sum_{j = 1}^{m - 1} \sum_{p = j + 1}^m a_j a_p $$ Then it follows that $$\forall a_1, \dots, a_{m + 1}, \ \big( \sum_{j = 1}^{m + 1} a_j \big)^2 = \sum_{j = 1}^{m + 1} a_j^2 + \sum_{j = 1}^{\underbrace{m + 1- 1}_m} \sum_{p = j + 1}^{m + 1} a_j a_p $$

Answer 2

Cross out all the $i$'s. They add nothing to the problem but extra ink. Then write out the identity for $m=2$ and $m=3$, without the summation signs and I think you'll see what's up right away.