Question Video: Finding the Common Ratio and General Term of a Geometric Sequence Mathematics

Video Transcript

In a geometric sequence, the ratio between any two successive terms is a fixed ratio 𝑟. Consider the sequence one-half, one-quarter, one-eighth, one sixteenth, and so on. Is this sequence geometric? Consider the sequence one-half, one-quarter, one-eighth, one sixteenth, and so on. What is the value of 𝑟? Consider the sequence one-half, one-quarter, one-eighth, one sixteenth, and so on. What is the general term of this sequence?

In this question, we’re given a reminder of what a geometric sequence is. It’s one which has a fixed ratio or common ratio between any two successive terms. In each of the three parts of this question, we’re considering the same sequence. And in the first part of this question, we’re asked if this given sequence is geometric. So let’s write down this sequence. If it is geometric, then there will be a common ratio 𝑟 between any two consecutive or successive terms. So let’s see if we can find a ratio between the first two terms, one-half and one-quarter. To find the ratio, we take the second term of one-quarter and divide it by one-half.

When we’re dividing fractions, we write the first fraction and we multiply it by the reciprocal of the second fraction. We can take out a common factor of two from the numerator and denominator. And then multiplying the numerators gives us one, and multiplying the denominators gives us two. That means that the ratio between the first and second term is one-half. Next, we will find the ratio between the second and third term of one-quarter and one-eighth, so we’ll calculate one-eighth divided by one-quarter. Multiplying by the reciprocal of the fraction one-quarter, we calculate one-eighth multiplied by four over one. And so, once again, we get the ratio of one-half.

It looks like we probably do have a common ratio, but it’s always worth checking all the terms to make sure that there is a common ratio on all of them. And when we calculate one sixteenth divided by one-eighth, we also get the ratio of one-half. Therefore, we have established that this sequence does have a fixed or common ratio, and so it must be geometric. And we can say yes as the answer for the first part of this question. In the second part of this question, we’re asked to find the value of 𝑟 for the same sequence. Remember that 𝑟 is the fixed ratio and it’s nice and simple. We have just worked it out to be one-half, which is the second part of this question answered.

In the final part of this question, we’re asked to find the general term of this sequence. And we can remember that the general term is another way of asking for the 𝑛th term. If we started with the first term written as 𝑎 sub one equal to one-half, then the second term would be 𝑎 sub two and it would be one-quarter. The third and fourth terms can be written as 𝑎 sub three and 𝑎 sub four. So when we’re finding the general term, we’re really looking for the rule that would allow us to work out the 𝑛th term or 𝑎 sub 𝑛.

We can remember that there is a general formula to allow us to work out the 𝑛th term of any geometric sequence. 𝑎 sub 𝑛 is equal to 𝑎 multiplied by 𝑟 to the power of 𝑛 minus one, where 𝑎 is the first term and 𝑟 is the common ratio. The values that we need to plug into the formula will be 𝑎 equals one-half, as that was the first term, and 𝑟 we worked out as one-half. Therefore, the 𝑛th term of this sequence can be given as a half multiplied by one-half to the power of 𝑛 minus one. When we give our answer, it’s important that we indicate the values of 𝑛. When we find the 𝑛th term or general term, we started with 𝑛 equals one. So the answer for the third part of the question is that the general term of the sequence is one-half multiplied by one-half to the power of 𝑛 minus one for values of 𝑛 greater than or equal to one.

We can, of course, further simplify this general term using one of the rules of exponents. If we consider that the first value of one-half is equivalent to one-half to the power of one, and so adding the exponents one and 𝑛 minus one will give us simply the exponent of 𝑛. If we then plugged in the values of 𝑛 equals one, two, three, or four into either of these formulas, we’d get the first four terms of the sequence that we were given, and so verifying that we have the correct answer for the general term.