Finding formulas for the variable "a" and summations

[Robb]

Homework Statement

Can someone explain how to get from step 5 to step 6. I'm not seeing the link. Thanks in advance!

Homework Equations

The Attempt at a Solution

 

[JulioHC]

Think about what the expression inside the brackets looks like, doesn't it remind you of an arithmetic series?

 

[Robb]

Does it relate to ∑ i = n(n+1)/2? Because the indexes of sigma begin at i =0 we use n-1 instead of n+1, in the right side of the equation, which would indicate i = 1 to n (for n+1, that is)? Except, adding from zero (i.e., 0+1+2+3+4, +...+n) and adding from n-1 (i.e. (n-1)+(n-2)+(n-3)+...+ 1), I get n+1 for the last column when the columns (of each respective summation) are summed.

 

[SammyS]

Your response indicates that you are quite confused.
Your first question seems to be:

Does it relate to ∑ i = n(n+1)/2? Because the indexes of sigma begin at i =0 we use n-1 instead of n+1, in the right side of the equation, which would indicate i = 1 to n (for n+1, that is)?

So, you seem to think that starting at i = 0 , rather than i = 1 somehow affects the final result?

Is 1 + 2 + 3 + ... + n any different than 0 + 1 + 2 + 3 + ... + n ?

Except, adding from zero (i.e., 0+1+2+3+4, +...+n) and adding from n-1 (i.e. (n-1)+(n-2)+(n-3)+...+ 1), I get n+1 for the last column when the columns (of each respective summation) are summed.

What columns are you referring to? I see no columns.

It seems that you know that $\ \displaystyle \sum_{i=1}^K i = \dfrac{K(K+1)}{2} \,.\$ Right?

So, rather than summing to K, or to n for that matter, are you not summing from 1 to (n−1) ?