Mathematical induction

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

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

Investigate Farey sequences of ratios of Fibonacci numbers.

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?

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

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

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

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

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?