Prove the formula for the sum of the squares of the first n positive integers

[doughishere]

in solution 1 why do we start with that specific sum ((1+i)^3-i^3)? is it just randomly picked or is there a reason that sum for this solution to example 5? is there a relation to the original sum we are asked to prove?

TIA.

 

[doughishere]

basically, just why do we pick this specific telescoping sum and not any other rando one?

 

[doughishere]

I think i figured it out. Is it fair to say that ever telescoping sum comes to the form An^3 + BN^2 + Cn + D and thus if we can write it in this form we can say its telescoping...thus we basically prove that both are "telescoping"(ie they bolth simplify to the form) and thus the formula works for our original sum...i think thats sloppy.

 

[doughishere]

test

\(\displaystyle 2x^{4}+5x^{2}-1\)


\(\displaystyle {-b\pm\sqrt{b^2-4ac}}{2a}\)

\int_{a}^{b} f(x)dx = F(b) - F(a)

\displaystyle{sin\left [\dfrac{x^3}{cos(x^3)}\right ]}

\(\displaystyle {\color{green} {\sum_{i=1}^n[(1+i)^{3}-(i)^{3}]}}\)

 

[doughishere]

merci beaucoup.