This article gives an introduction to mathematical induction, a powerful method of mathematical proof.

Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.

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

Find the smallest value for which a particular sequence is greater than a googol.

Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.

Find the sum, f(n), of the first n terms of the sequence: 0, 1, 1, 2, 2, 3, 3........p, p, p +1, p + 1,..... Prove that f(a + b) - f(a - b) = ab.

Yatir from Israel describes his method for summing a series of triangle numbers.

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

A story for students about adding powers of integers - with a festive twist.

Watch the video to see how to sum the sequence. Can you adapt the method to sum other sequences?

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

Prove that k.k! = (k+1)! - k! and sum the series 1.1! + 2.2! + 3.3! +...+n.n!

Prove that the sum of the reciprocals of the first n triangular numbers gets closer and closer to 2 as n grows.

What is the sum of: 6 + 66 + 666 + 6666 ............+ 666666666...6 where there are n sixes in the last term?

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

This article by Alex Goodwin, age 18 of Madras College, St Andrews describes how to find the sum of 1 + 22 + 333 + 4444 + ... to n terms.

Watch the video to see how Charlie works out the sum. Can you adapt his method?

Find relationships between the polynomials a, b and c which are polynomials in n giving the sums of the first n natural numbers, squares and cubes respectively.

Watch the video to see how to add together an arithmetic sequence of numbers efficiently.

Can you see how this picture illustrates the formula for the sum of the first six cube numbers?

This is an interactivity in which you have to sort into the correct order the steps in the proof of the formula for the sum of a geometric series.

By tossing a coin one of three princes is chosen to be the next King of Randomia. Does each prince have an equal chance of taking the throne?

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?

Is it true that a large integer m can be taken such that: 1 + 1/2 + 1/3 + ... +1/m > 100 ?

Each week a company produces X units and sells p per cent of its stock. How should the company plan its warehouse space?