Question Video: Writing a Recursive Formula for a Sequence Mathematics

Video Transcript

Consider the sequence four, 10, 22, 46, and so on. Which of the following recursive formulas can be used to calculate successive terms of the sequence for an index 𝑛 is greater than or equal to one? Option (A), 𝑎 sub one equals four, 𝑎 sub 𝑛 plus one equals 𝑎 sub 𝑛 plus six. Option (B), 𝑎 sub one equals four, 𝑎 sub 𝑛 plus one equals two 𝑎 sub 𝑛. Option (C), 𝑎 sub one equals four, 𝑎 sub 𝑛 plus one equals five over two 𝑎 sub 𝑛. Or option (D), 𝑎 sub one equals four, 𝑎 sub 𝑛 plus one equals two 𝑎 sub 𝑛 plus two.

Let’s begin this question by considering we are told that the sequence four, 10, 22, 46, and so on is going to be defined by a recursive formula. A recursive formula is a formula in which the terms of the sequence are defined using one or more of the previous terms. So let’s think about how we define the terms of a sequence given an index value of 𝑛. Here, we are told that 𝑛 is greater than or equal to one. So that means that we can say that the first term would be 𝑎 sub one. The second term can be denoted as 𝑎 sub two, the third term as 𝑎 sub three, and so on. If we are given a recursive formula and we need to generate the sequence, we must always be given enough information so that we can start the sequence. Usually, this is the value of the first term.

If we look at the available answer options, in each case, we are told that the value of 𝑎 sub one is equal to four. And this would match with the given sequence which has a first term of four. So this information alone wouldn’t be enough to eliminate any of the given answer options. What we can do then is take each answer option in turn and generate a sequence given its recursive formula to see if it will match our given sequence.

Let’s look at answer option (A). And we know that we are told that the first term in this sequence is four because that’s 𝑎 sub one. Next, to find the second term 𝑎 sub two, this will be equal to 𝑎 sub one plus six. And since we’ve worked out or we’re given in this case that 𝑎 sub one is equal to four, then 𝑎 sub two is equal to four plus six, which is 10. If we check the given sequence at this point, the first term is four and the second term is 10. So let’s continue to work out the next term of this sequence. Since the next term is 𝑎 sub three, this can be found by 𝑎 sub two plus six. We’ve already worked out that 𝑎 sub two is equal to 10, so 10 plus six would give us 16. However, if we then check this third term with the third term of the given sequence, we can see that these do not match. This means that the recursive formula given in answer option (A) does not match the given sequence, and so it must be incorrect.

We can then perform the same method in answer option (B), beginning once again with a first term 𝑎 sub one of four. This time, the recursive part of the formula tells us that to find a term in the sequence, we take the term before it and double it. So 𝑎 sub two is equal to two times 𝑎 sub one. We know that 𝑎 sub one is equal to four, so 𝑎 sub two is equal to two times four, which is eight. However, if we look at the given sequence, although 𝑎 sub one is four, 𝑎 sub two was given as 10. And these two second terms do not match. So we can eliminate answer option (B).

Continuing then with answer option (C), this time the recursive part of the formula tells us that to find a given term in the sequence, we take the previous term and multiply it by five over two. To calculate the second term then, we take the first term of four and multiply it by five over two which gives us 20 over two. And that can be simplified to 10. So far, these two terms match the given sequence. So let’s see if we can calculate the third term. 𝑎 sub three is equal to five over two times 𝑎 sub two. And five over two times 10 is 50 over two and that’s equal to 25. But of course, we can see that in the given sequence, the third term is 22 and not 25. So therefore, we eliminate answer option (C).

Therefore, we have one answer remaining, answer option (D). So let’s check the sequence that this generates. This time, the recursive part of the formula tells us that to find 𝑎 sub 𝑛 plus one, we have two times 𝑎 sub 𝑛, that’s the term before, plus two. 𝑎 sub two is therefore equal to two times 𝑎 sub one which is four plus two. Two times four is eight plus two is equal to 10. 𝑎 sub three will be equal to two times 10 plus two which is 22. And we have the first three terms here matching the given sequence. So let’s check the fourth term. When we multiply 22 by two and add two, we do indeed get a fourth term of 46. Hence, we have the answer that the given sequence can be described as the recursive formula 𝑎 sub one equals four, 𝑎 sub 𝑛 plus one equals two 𝑎 sub 𝑛 plus two.

But before we finish with this question, let’s see if we could have completed this question without the possible answer options. Let’s say that we just have the sequence and we need to describe it using a recursive formula. The first thing we might ask ourselves is, how could we describe this sequence in words? And in fact, we might notice that to find any term in this sequence, we double the previous term and add two. Therefore, to write a recursive formula, we would start by saying 𝑎 sub 𝑛 plus one that means to find any term. Then, the previous term would be defined as 𝑎 sub 𝑛. And we know that we double that. We then add two to this value. So we would have that 𝑎 sub 𝑛 plus one is equal to two 𝑎 sub 𝑛 plus two.

But remember, if we want to generate a sequence from a recursive formula, we need a starting value. In this case, we would need to describe to anyone else who wanted to generate the sequence that the first term is equal to four. So 𝑎 sub one is equal to four. And this is exactly what we had in answer option (D). So either method could be used to determine the recursive formula for this given sequence.