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. Is this sequence geometric? Consider the sequence one-half, one-quarter, one-eighth, one-sixteenth. What is the value of 𝑟? Consider the sequence one-half, one-quarter, one-eighth, one-sixteenth. What is the general term of this sequence?
In this question, we’re asked to consider three different things about the same sequence. In the first part of this question, we’re asked if this is a geometric sequence. And we’re also given a handy reminder of what a geometric sequence is. It’s one where the ratio between any two successive terms is a fixed ratio.
So let’s consider the terms that we’re given in this sequence: one-half, one-quarter, one-eighth, and one-sixteenth. Because we know that there’ll be a fixed ratio in a geometric sequence, we’re really asking what must we multiply any term in the sequence by to get the successive term. Notice that this is different in arithmetic sequence because in an arithmetic sequence we’re adding or subtracting between terms.
What we might notice first of all is that we can divide one-half by two to get one-quarter. We could also divide one-quarter by two to get one-eighth. And one-eighth divided by two would also give us the next term, one-sixteenth. However, we need to present our ratio in terms of the number that we’re multiplying by. So how do we change division by two into something that we’re multiplying by?
Well, we can write it as multiplying by the fraction one-half. And so we’ve discovered that if we take any term in this sequence and multiply it by one-half, we get the next term in the sequence. Since this is exactly what happens in a geometric sequence, then we can give the answer for the first part of this question as yes, since we have a geometric sequence.
In the second part of this question, we’re asked for the value of 𝑟. Remember that 𝑟 is the ratio between terms. And so we’re looking for the value that we multiply any term by to get the next. It’s the value of one-half that we’ve already worked out. And so this is the second part of the question answered.
In the final part of this question, we need to work out the general term of the sequence. We can recall that, for any geometric sequence, where the first term is written as 𝑎, the fixed ratio is given as 𝑟, we can find any term with index 𝑛, 𝑎 sub 𝑛 as 𝑎 times 𝑟 to the power of 𝑛 minus one. This comes from the fact that if we denote the first term as 𝑎, then we know that the second term would be the first term multiplied by the fixed ratio 𝑟. So that would give us 𝑎 times 𝑟. And then the third term would be the second term of 𝑎𝑟 multiplied by another 𝑟.
So let’s consider how we would write the general or 𝑛th term for the given sequence. Well, we established in the second part of this question that the ratio 𝑟 between terms is one-half. The value of 𝑎 is simply the first term in the sequence, which is also one-half. When we write the general term, we’re going to say that, for any index 𝑛, 𝑎 sub 𝑛 is equal to 𝑎, that’s one-half, multiplied by 𝑟, which is one-half, to the power of 𝑛 minus one. And so that’s the general term of this sequence.
However, we need to also note what values that 𝑛 can take. This sequence formula will work for values starting with 𝑛 is equal to one. And so alongside the general term then, we must note that the index 𝑛 is greater than or equal to one. And so we can give the answer that the general term of this sequence is 𝑎 sub 𝑛 is equal to one-half times one-half to the power of 𝑛 minus one for values 𝑛 is greater than or equal to one.
We could also apply the laws of exponents to simplify this general term a little more. We can remember that if we have a value 𝑎 to the power of 𝑥 multiplied by 𝑎 to the power of 𝑦, then this can be simplified as 𝑎 to the power of 𝑥 plus 𝑦. When we look at what we have here in this general term, one-half multiplied by one-half to the power of 𝑛 minus one could also be written as one-half to the power of one multiplied by one-half to the power of 𝑛 minus one. And then when we add the exponents of one and 𝑛 minus one, we get a value of 𝑛. We could therefore give either of these formulas for 𝑎 sub 𝑛 as the answer to the general term of this sequence.
