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Broad Topics > Using, Applying and Reasoning about Mathematics > Mathematical induction

Tens

Age 16 to 18 Challenge Level:

When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?

Converging Product

Age 16 to 18 Challenge Level:

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)... ?

Particularly General

Age 16 to 18 Challenge Level:

By proving these particular identities, prove the existence of general cases.

Farey Fibonacci

Age 16 to 18 Short Challenge Level:

Investigate Farey sequences of ratios of Fibonacci numbers.

Farey Neighbours

Age 16 to 18 Challenge Level:

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Symmetric Tangles

Age 14 to 16

The tangles created by the twists and turns of the Conway rope trick are surprisingly symmetrical. Here's why!

Elevens

Age 16 to 18 Challenge Level:

Add powers of 3 and powers of 7 and get multiples of 11.

An Introduction to Mathematical Induction

Age 16 to 18

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

Fibonacci Fashion

Age 16 to 18 Challenge Level:

What have Fibonacci numbers to do with solutions of the quadratic equation x^2 - x - 1 = 0 ?

Golden Fractions

Age 16 to 18 Challenge Level:

Find the link between a sequence of continued fractions and the ratio of succesive Fibonacci numbers.

Gosh Cosh

Age 16 to 18 Challenge Level:

Explore the hyperbolic functions sinh and cosh using what you know about the exponential function.

Water Pistols

Age 16 to 18 Challenge Level:

With n people anywhere in a field each shoots a water pistol at the nearest person. In general who gets wet? What difference does it make if n is odd or even?

Obviously?

Age 14 to 18 Challenge Level:

Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.

Dirisibly Yours

Age 16 to 18 Challenge Level:

Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

Counting Binary Ops

Age 14 to 16 Challenge Level:

How many ways can the terms in an ordered list be combined by repeating a single binary operation. Show that for 4 terms there are 5 cases and find the number of cases for 5 terms and 6 terms.

Age 14 to 16 Challenge Level:

A walk is made up of diagonal steps from left to right, starting at the origin and ending on the x-axis. How many paths are there for 4 steps, for 6 steps, for 8 steps?

Age 7 to 18 Challenge Level:

Libby Jared helped to set up NRICH and this is one of her favourite problems. It's a problem suitable for a wide age range and best tackled practically.

Golden Powers

Age 16 to 18 Challenge Level:

You add 1 to the golden ratio to get its square. How do you find higher powers?

Binary Squares

Age 16 to 18 Challenge Level:

If a number N is expressed in binary by using only 'ones,' what can you say about its square (in binary)?

Growing

Age 16 to 18 Challenge Level:

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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?

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?