How do I do a proof by induction?

For this explanation we will use the following example from a 2013 exam paper:

   If u₁= 2 and u_n+1=(5u_n-3)/(3u_n-1), then prove that u_n=(3n+1)/(3n-1) for all n>=1

The first step of any proof by induction is to make the assumption that what we want to prove is true for a particular value n = k:

  Assume there exists k such that u_k=(3k+1)/(3k-1)

We must then prove that it is also true for n = (k+1), we start by finding u_k+1 using the original formula:

  u_k+1 = (5u_k-3)/(3u_k-1) = (5*(3k+1)/(3k-1) - 3)/(3*(3k+1)/(3k-1) - 1) = ... = (3k+4)/(3k+2)

We now want to write this in terms of k+1, in this case it is fairly straightforward but other times it may be harder to see:

  u_k+1 = (3k+4)/(3k+2) = (3(k+1) - 3 + 4)/(3(k+1) - 3 +2) = (3(k+1)+1)/(3(k+1)-1)

When written in terms of k+1, u_k+1 should now be in the form that we want to prove for u_nor a form that can be rearranged into that one. There is still one step left however which is CRUCIAL for this to be a proper proof by induction. We have to prove this is true for a certain value of n, in this case n = 1:

   u₁ = 2 = (31+1)/(31-1) therefore the assumption is true for n = 1. It is therefore true for n = 1, 2, 3, ...

This last step is usually very simple but can often be overlooked so make sure to include it!