- #1
steelphantom
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Homework Statement
Prove 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all natural numbers n.
Homework Equations
The Attempt at a Solution
Well, this seems like the typical induction proof, so I start by testing the hypothesis at 1: 1^3 = 1^2 = 1. Then I assume that the equation is true for n, and try to prove it true for n + 1.
1 ^3 + 2^3 + ... + n^3 + (n + 1)^3 = (1 + 2 + ... + n)^2 + (n + 1)^3
I have this, but how do I get it to be (1 + 2 + ... + n + n + 1)^2? Is it some crazy factorization that I'm overlooking? It seems pretty easy.
Homework Statement
Prove that 7^n - 6n - 1 is divisible by 36 for all positive integers n.
Homework Equations
The Attempt at a Solution
Well, the statement seems to be saying the following: 7^n - 6n - 1 = 36k, where k is a positive integer. As with the first problem, testing it with 1 works. Assuming it is true for n + 1 would mean the following, if I'm correct:
7^(n + 1) - 6(n + 1) - 1 = 36j, where j is a positive integer. At this point, I'm not sure how to proceed.
I know I'm probably overlooking some obvious stuff here, so a little hint is probably all I'll need to finish these problems off. Thanks! :)