totally confused!?

Question

Can anybody please help me show that

n
Σi³ = (n²(n+1)²) / 4
i=1

Where n is any natural number, N.

Thank you so much! =]

Answer 1

(r+1)^4 - r^4 = 4r^3 +6r^2+4r+1
sum for r=1 to n noting lhs terms that cancel out
(n+1)^4 - 1^4 = 4 sum r^3 +6 sum r^2 + 4sum r +n
you know sum r and sum r^2 (or can derive them same way)
so you can juggle round to get answer.

Or prove by induction. Assume true for n
sum [1,n+1] = sum[1,n] + (n+1)^3
= [ n^2(n+1)^2 +4(n+1)^3 ] /4
= [ n^2 + 4n + 1 ] (n+1)^2/4
= (n+2)^2 (n+1)^2 /4 i.e. also true for n+1
sum [1,1] = 1^3 = 1^2 2^2 / 4
therefore true for all positive n

Answer 2

Suppose it is proved for n = k. Then prove it for n = k+1 .
If u do so, cos' n =1 satisfies it, u can say it is proved for n = 2 , n = 3 , ...