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.

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

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

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

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

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.

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

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

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

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.

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

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

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

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.

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.

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

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

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

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

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?

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

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

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

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?