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.

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?

Investigate Farey sequences of ratios of Fibonacci numbers.

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?

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

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