# Help with mathematical assertions for natural numbers

## Homework Statement

Prove by Mathematical Induction that the assertion,
n
∑ r^2 = n/6 (n+1)(2n+1)
r=1

holds for every natural number n.

## Homework Equations

Ok, so basically, how do you solve this question? I have got to the Induction step but I'm not sure how to do it.

## The Attempt at a Solution

I've replaced n with k so I have,

1^2 + 2^2 + 3^2 + ... k^2 = k/6 (k+1)(2k+1)

Then I've added the (k+1)th term to each side to I have,

1^2 + 2^2 + 3^2 + ... k^2 + (k+1)^2 = k/6 (k+1)(2k+1) + (k+1)^2

So where do I go from here?

## Answers and Replies

I like Serena
Homework Helper
Hi MegaDeth! Substitute (k+1) in n/6 (n+1)(2n+1) and check if that is equal to your expression.

Sorry, but I'm not really sure how to do that :S How do I substitute it and equal it to the expression?

To complete the proof, you have to show that the right side of the equation is (k+1)/6 ((k+1)+1)(2(k+1)+1). All it requires is algebra to re-express the right side in a way that shows your assertion is correct.

I like Serena
Homework Helper
Sorry, but I'm not really sure how to do that :S How do I substitute it and equal it to the expression?

If you substitute n=(k+1) in n/6 (n+1)(2n+1), you get:

(k+1)/6 ((k+1)+1)(2(k+1)+1).

Is that the same as k/6 (k+1)(2k+1) + (k+1)^2?