Challenge Level

Why do this problem?

The problem gives step by step guidance so that learners only need to apply what they know about the Binomial expansion of $(k+1)^n$ and do some simple algebraic manipulation to be able to find general formulae for the sums of powers of the integers. As the name suggests the method makes use of the 'telescoping' property so that all the intermediate terms disappear leaving only the first and last.

Possible
Approach

You might choose to introduce this method just for the sum of
the squares of the integers as pictured in the pyramid
illustration.

Key question

What is $[2^2-1^2] + [3^2-2^2] +
[4^2-3^2] + \cdots + [(n + 1)^2-n^2]$?

Possible support

Try the problems
Natural Sum,
More Sequences and Series, and
OK Now Prove It
Possible support

Read the article
Proof by Induction.

Possible extension

Seriesly