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By sketching a graph of a continuous increasing function, can you prove a useful result about integrals?

Can you work through these direct proofs, using our interactive proof sorters?

In the limit you get the sum of an infinite geometric series. What about an infinite product (1+x)(1+x^2)(1+x^4)... ?

Investigate the molecular masses in this sequence of molecules and deduce which molecule has been analysed in the mass spectrometer.

To find the integral of a polynomial, evaluate it at some special points and add multiples of these values.

Decipher a simple code based on the rule C=7P+17 (mod 26) where C is the code for the letter P from the alphabet. Rearrange the formula and use the inverse to decipher automatically.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?

Can you find a way to prove the trig identities using a diagram?

Given any two polynomials in a single variable it is always possible to eliminate the variable and obtain a formula showing the relationship between the two polynomials. Try this one.

The nth term of a sequence is given by the formula n^3 + 11n . Find the first four terms of the sequence given by this formula and the first term of the sequence which is bigger than one million. . . .

A preview of some of the beam deflection mechanics you will look at in the first year of an engineering degree

What have Fibonacci numbers got to do with Pythagorean triples?

An account of multiplication of vectors, both scalar products and vector products.

A space craft is ten thousand kilometres from the centre of the Earth moving away at 10 km per second. At what distance will it have half that speed?

Can you interpret this algorithm to determine the day on which you were born?

Get to grips with the variables behind the motion of a simple pendulum

Discover how Heron of Alexandria missed his chance to explore the unknown mathematical land of complex numbers.

A sequence of numbers x1, x2, x3, ... starts with x1 = 2, and, if you know any term xn, you can find the next term xn+1 using the formula: xn+1 = (xn + 3/xn)/2 . Calculate the first six terms of this. . . .

Be playful with graphs and networks, and see what theorems you can discover!

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different?. . . .

Derive Euler's buckling formula from first principles.

An introduction to the binomial coefficient, and exploration of some of the formulae it satisfies.

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

See how enormously large quantities can cancel out to give a good approximation to the factorial function.

The harmonic triangle is built from fractions with unit numerators using a rule very similar to Pascal's triangle.

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum of two or more cubes.

We continue the discussion given in Euclid's Algorithm I, and here we shall discover when an equation of the form ax+by=c has no solutions, and when it has infinitely many solutions.

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.

Take a complicated fraction with the product of five quartics top and bottom and reduce this to a whole number. This is a numerical example involving some clever algebra.

You can differentiate and integrate n times but what if n is not a whole number? This generalisation of calculus was introduced and discussed on askNRICH by some school students.

Here explore some ideas of how the definitions and methods of calculus change if you integrate or differentiate n times when n is not a whole number.

Relate these algebraic expressions to geometrical diagrams.

Some statements which can be proved using induction, and some example proofs.

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

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

Selection of STEP Statistics questions with hints for students to undertake as part of the Statistics STEP Prep Module.

Some simple ideas about graph theory with a discussion of a proof of Euler's formula relating the numbers of vertces, edges and faces of a graph.

Here is a proof of Euler's formula in the plane and on a sphere together with projects to explore cases of the formula for a polygon with holes, for the torus and other solids with holes and the. . . .

Some of our experiences of discovering and using triangle numbers in a range of contexts.

A polite number can be written as the sum of two or more consecutive positive integers. Find the consecutive sums giving the polite numbers 544 and 424. What characterizes impolite numbers?

What is an AC voltage? How much power does an AC power source supply?

Crack this code which depends on taking pairs of letters and using two simultaneous relations and modulus arithmetic to encode the message.

Infinity is not a number, and trying to treat it as one tends to be a pretty bad idea. At best you're likely to come away with a headache, at worse the firm belief that 1 = 0. This article discusses. . . .

10 of our best problems to help you prepare to study chemistry at university.

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.