Mathematical induction

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

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)... ?

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.

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?

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

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

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

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?

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