[The narrator reads out the onscreen text.]

NARRATOR: Question four. Here are four diagrams made of matches.

[Four diagrams are each made up of a row of squares made up of matches, their sides touching. The first diagram consists of one square, the second of two squares, the third of four squares and the fourth of eight squares. Each square has a diagonal line in the centre.]

NARRATOR: Part a. Write down a sequence of six numbers giving the number of matches in each of these diagrams and the next two in the obvious sequence. Part b. Give the rule for finding the next term from the previous term.

Part a. In the first diagram, we can see that there is 5 matches. In the second diagram, there is 9 matches. In the third diagram, there is 17 matches. And in the fourth diagram, there is 33 matches.

We can see that each diagram is created by doubling the number of squares from the previous diagram. So the first diagram contains one square, the second diagram two squares, the third diagram four squares and the fourth diagram eight squares. But this isn't equivalent to simply multiplying the number of matches by two. And that's because when we add on an additional square, we're not adding on the match in the middle. Similarly, when we add on two squares to double the number of squares, we're not adding on the match in the middle. So each time we are doubling the number of matches but subtracting one in order to account for that middle match that's not being added.

Let's check whether that works. So when we double 5, 5 times 2 gives 10, but subtracting 1 gives 9. 9 times 2 gives 18, subtract 1 is 17. 17 times 2 is 34, subtract 1 is 33. Continuing this same pattern, 33 times 2 is 66, subtract 1 is 65. And 65 times 2 is 130, subtracting 1 to give 129. So the sequence of six numbers is 5, 9, 17, 33, 65 and 129.

Part b. Give the rule for finding the next term from the previous term. We've established the sequence to be 5, 9, 17, 33, 65 and 129, and have noted that we obtained each number by multiplying the previous term by two, and then subtracting 1. For example, Term two is found by multiplying Term one by 2 and subtracting 1. Term three is found by multiplying Term two by 2 and subtracting 1. Term four is found by multiplying Term two by three and subtracting 1. And we can continue on.

So generally then, to find Term n - the nth numbered term - we would multiply the previous term by 2. So 2 times Term (n minus 1) - the term before Term n - and then subtract one.

[2 times (Term n - 1) – 1]