Question Video: Determining Whether a Given Sequence Is Arithmetic or Geometric

Difference between Arithmetic and Geometric Sequence
Difference between Arithmetic and Geometric Sequence

Video Transcript

Is the following sequence arithmetic or geometric? 11, 33, 99, 297, and the sequence continues.

Let’s begin by recalling what we mean by arithmetic and geometric sequences. A sequence is arithmetic if there is a common difference between the terms. This means we always add or subtract the same amount to get from one term to the next. For example, the sequence 10, seven, four, one, negative two is an example of an arithmetic sequence, because to get from one term to the next, we always subtract three. Here, we would say that the common difference of the sequence, which we denote using the letter 𝑑, is negative three because we are subtracting three to get from one term to the next.

On the other hand, a sequence is geometric if there is a common ratio between the terms. This means that we always multiply by the same amount to get from one term to the next. For example, the sequence five, 10, 20, 40, 80, and so on is an example of a geometric sequence, because to get from one term to the next, we always multiply by the same amount, which is two. This time, we would say that the common ratio of the sequence, which we denote using the letter 𝑟, is two. The common ratio can also be a fraction, which would be equivalent to always dividing by the same amount. And it can also be negative.

Let’s now look at the sequence we’ve been given. First, we’ll consider whether it is arithmetic. And to do this, we’ll look at the differences between each pair of terms. We can label the terms as 𝑎 sub one, 𝑎 sub two, 𝑎 sub three, 𝑎 sub four, and so on. Finding the first difference, 𝑎 sub two minus 𝑎 sub one — that’s the second term minus the first — we have 33 minus 11, which is 22. Finding the difference between the next pair of successive terms, 𝑎 sub three minus 𝑎 sub two, gives 99 minus 33, which is 66.

Now, we can already see that these differences are not constant, and so the sequence isn’t arithmetic. But let’s just check the final pair of terms to confirm this. 𝑎 sub four minus 𝑎 sub three is 297 minus 99, which is 198. So, as we’ve already seen, the difference between each pair of successive terms is not constant. And so the sequence is not arithmetic.

Let’s now consider whether the sequence is geometric. Just because it isn’t arithmetic, it doesn’t automatically mean that it will be geometric. There are lots of other types of sequences that are neither arithmetic nor geometric. To find the ratio between each pair of successive terms, we divide each term by the previous one. So, we start with 𝑎 sub two divided by 𝑎 sub one; that’s 33 divided by 11, which is three. Then, the next pair of terms 𝑎 sub three divided by 𝑎 sub two, that’s 99 divided by 33, which is also three. It’s looking likely that the sequence is geometric, but we need to check the final pair of terms. 𝑎 sub four divided by 𝑎 sub three, that’s 297 divided by 99, is also equal to three.

We’ve found that this sequence does indeed have a common ratio. So, we can answer that the sequence 11, 33, 99, 297, and so on is a geometric sequence.

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