Question Video: Finding the General Term to Work Out Terms in a Sequence Mathematics

Video Transcript

If 𝑎 sub 𝑛 is a sequence defined as 𝑎 sub one equals 11 and 𝑎 sub 𝑛 plus one equals 𝑎 sub 𝑛 minus three, where 𝑛 is greater than or equal to one, then the fourth term equals what.

We’re given four answer options: two, four, five, or eight. In this question, we’re given a formula for a sequence. This type of formula is called a recursive formula. And that’s when the terms of a sequence are defined using one or more previous terms. If we wanted to describe this term in words, we would say that for any term with index 𝑛 plus one, we take the term before it — that’s the one with index 𝑛 — and we subtract three. And so if we wanted to find the fourth term — that’s the term with index four — that means that 𝑛 plus one must be equal to four, and so 𝑛 must be three. And so the fourth term must be equal to the third term minus three. But how do we find the third term?

Well, the third term — that’s the term with index three — must happen when 𝑛 plus one is three. And so 𝑛 must be equal to two. So the third term is equal to the second term minus three. Of course, we don’t know the second term either. But you’ve guessed it! It’s going to be the first term minus three. And this is also one of the disadvantages of recursive formulas because we need to work out every term up to the term that we need.

We do get a little bit of relief here because we’re actually given the first term. 𝑎 sub one is equal to 11. So now we can work forwards through the sequence. If 𝑎 sub one is equal to 11 and 𝑎 sub two is equal to 𝑎 sub one minus three, then 𝑎 sub two, the second term, is equal to 11 minus three. And that’s equal to eight. As the third term is equal to the second term minus three, then our third term must be equal to eight minus three, which is five. And finally then, the fourth term is the third term minus three. And so five minus three is equal to two. We can therefore give the answer that the fourth term of the sequence is that given in option (A). It’s the term two.