Thinking Forwards and Backwards in Math

An exam skill I wish I had been better at

Many of you who have taken math exams in the past will probably be familiar with the type of question where you are given a result and you have to derive it or prove it. These types of questions are very common in upper high school and university math papers.

When I was younger my approach to these would always be the same — I would think forwards. What I mean by this is that I would start with the facts given to me and then try to work forwards to derive the result I was asked to derive. If I wasn’t getting to that result and was running out of time I’d often just give up and move onto another question.

But I’ve learned nowadays that for problems like these you need to think in two directions. If you think forwards from the information you are given, and backwards from the result you have to derive, then you will increase your chances of seeing the critical insight you need to complete the problem, which should lie at the intersection of both directions of thinking.

This is best described using an example. Here is a question from a 1988 UK examination for advanced high school students.

Attacking this problem

Let’s take the first part first. Thinking purely forwards, I see that I have a function with a product of two linear expressions in the denominator. So I probably have to do some manipulation of this function to express it in a more helpful form. But it’s not immediately clear what kind of manipulation might be helpful.

However, thinking backwards, and looking at the first result I am being asked to derive, I notice two things:

• The result is composed of two infinite geometric series.
• The result only holds for absolute values of x less than 1. Or alternatively -1 < x < 1.

The second point is a pretty common condition for convergence of an infinite geometric series in x, so it’s likely that this problem is about the sum of infinite series. And given that our result contains two…