How Do i solve this mathematical induction question ? :)?

Question

For any n >= 1, 1^3 + 2^3 + 3^3 + ... + n^3 = (1 + 2 + 3 + ... + n)^2.

Answer 1

If you are allowed to use the well known result that
1 + 2 + 3+ 4 . . . + n = n(n + 1)/2 then the whole thing becomes easier. We then want to prove that
1^3 + 2^3 +3^3 . . . + n^3 = (n(n + 1)/2)^2 = n^2(n + 1)^2/4
Check that it works with n = 1 and it does.
Assume true for n = k then
1^3 + 2^3 + 3^3 . . . + k^3 + (k + 1)^3
= k^2(k + 1)^2/4 + (k + 1)^3 = (k + 1)^2(k^2/4 + k + 1)
=(k + 1)^2(k^2 + 4k + 4)/4 = (k + 1)^2(k + 2)^2/4
This is the formula for n = k + 1 so if true for n = k it is also true for n = k + 1. Hence true for all n by induction.

Answer 2

1. check it for n =1 1^3 =1, 1^2 = 1 - TRUE.
2. assume it is correct for n, show it is correct for n+1
1 + .. + n = k
1^3+...+(n+1)^3 =? (1+...(n+1))^2
k^2 + (n+1)^3 ?= (k + (n+1))^2 = k^2 + 2K(n+1) + (n+1)^2
(n+1)^3 ?= (2k+ (n+1))*(n+1)
(n+1)^2 ?= 2k + n + 1
n^2 + 2n + 1 ?= 2k + n + 1
k ?= (n^2 + n)/2 -> true, so if it is true for 1 and for any n this is true for n+1, according to induction rule it is true for all positive numbers

Answer 3

I always feel bad when I see math questions. I think I have a mental block when it comes to math. I know someone out there can help you.. I wont get the best answer but at least I vented my feelings about my desire to want to know math.