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

Challenge Level

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

Challenge Level

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

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

Challenge Level

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

Challenge Level

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

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

Challenge Level

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

Challenge Level

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.

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.

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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

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

Challenge Level

Can you use the given image to say something about the sum of an infinite series?

Challenge Level

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

Challenge Level

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.

Challenge Level

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

Challenge Level

Can you correctly order the steps in the proof of the formula for the sum of the first n terms in a geometric sequence?

Challenge Level

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?

Challenge Level

Discover a way to sum square numbers by building cuboids from small cubes. Can you picture how the sequence will grow?

Challenge Level

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

Challenge Level

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

Challenge Level

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

Challenge Level

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?