Suppose that 3=2/x(1)=x(1)+(2/x(2))=x(2)+(2/x(3))=x(3)+(2/x(4))+...Guess an expression, in terms of n, for x(n). Then, by induction or otherwise, prove the correctness of your guess.
We have 3=2/x1=x1+2/x2=x2+2/x3=x3+2/x4=.... How can we guess at a general expression? One way is to try and see one by looking at the first few terms and looking for a pattern. We can calculate straight away x1=2/3, and then from 3=x1+2/x2, as we now know x1 we rearrange to find x2=6/7. Using the general form of 3=xn+2/xn+1, we can find the next few terms as 14/15, 30/31, 62/63... Patterns that we can spot: the numerator is always one less than the denominator, the denominator is always odd and the numerator of each term is double the denominator of the previous. This repeated doubling should point you in the direction of powers of 2. Some testing out with powers of 2 gives us the general expression of xn=(2n+1-2)/(2n+1-1).Now it remains to prove this by induction, which consists of three parts: base case, inductive step and conclusion. Base case we have already done but it doesn't hurt to restate x1=2/3. For the inductive step, we can use that 3=xn+2/xn+1 for n>0. Assume that the formula above holds for n, and then show that it holds for n+1. Then in the conclusion state "by the principle of mathematical induction", and clearly state that we have proved the general expression for xn in terms of n.
