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.

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.

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.

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

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 ?