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Degree Ceremony

What does Pythagoras' Theorem tell you about these angles: 90°, (45+x)° and (45-x)° in a triangle?

OK! Now Prove It

Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?

Overarch 2

Bricks are 20cm long and 10cm high. How high could an arch be built without mortar on a flat horizontal surface, to overhang by 1 metre? How big an overhang is it possible to make like this?

Telescoping Series

Age 16 to 18
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
Read the article Proof by Induction.

Possible extension