problem Summats Clear Age 16 to 18 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.
problem Overarch 2 Age 16 to 18 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?
problem OK! Now prove it Age 16 to 18 Challenge level Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?
problem Degree Ceremony Age 16 to 18 Challenge level Can you find the sum of the squared sine values?
problem Telescoping series Age 16 to 18 Challenge level Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.
article Sums of Powers - A Festive Story A story for students about adding powers of integers - with a festive twist.
article An introduction to mathematical induction This article gives an introduction to mathematical induction, a powerful method of mathematical proof.
article The kth sum of n numbers Yatir from Israel describes his method for summing a series of triangle numbers.
article Powerful properties Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
article Sum the Series 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.