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

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

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.

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

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

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

Can you correctly order the steps in the proof of the formula for the sum of a geometric series?

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

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

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?

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

How can visual patterns be used to prove sums of series?

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

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!

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?

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

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

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!

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

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

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

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

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?

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