Video Transcript
What is the order of the term in the geometric sequence six, 24, 96, and so on whose value is 1,572,864?
We can remember that a geometric sequence is a sequence which has a fixed ratio between successive or consecutive terms. Here, we need to work out the order or position of the term that has this value of 1,572,864. In order to help us with this, we can work out the 𝑛th term of this geometric sequence. And we should remember the formula to do this, which gives us that 𝑎 sub 𝑛 is equal to 𝑎 times 𝑟 to the power of 𝑛 minus one. The value of 𝑎 is sometimes given as 𝑎 sub one. And that represents the first term of the sequence. The value of 𝑟 is the ratio between terms.
Given that we have the first three terms of the sequence, then we can work out the ratio 𝑟 and find the first value. In order to find the ratio, we can take any term and divide it by the term that comes immediately before it. So using the first and second term, if we take 24 and divide it by six, we get the value of four. In other words, six multiplied by four would give us 24. The same would be true if we used the second and third terms. We can see that the ratio between the terms is four.
And so we know that 𝑟 is equal to four. When it comes to the first term, well, we’re given here that the first term is six. So that means that 𝑎 is equal to six. We can then fill in the values of 𝑎 and 𝑟 into this general term formula, remembering that 𝑟 is the fixed ratio. This gives us 𝑎 sub 𝑛 equals six times four to the power of 𝑛 minus one. When we have a general or 𝑛th term of a sequence, we could use that to find the value of any term in this sequence. In this question, however, we’re given the value of a term. So we could say that this is the value of 𝑎 sub 𝑛. We can then use the general term to work out what value of 𝑛 would produce this value.
And therefore, we can substitute this value for 𝑎 sub 𝑛 into the general term. Dividing through by six gives us that 262,144 is equal to four to the power of 𝑛 minus one. There are a number of different ways in which we could then solve to find this exponent of four. One way is to use a trial and improvement method, where we find whatever value it is that gives us 262,144. We could then go from there to find the value of 𝑛.
An alternative method is to take logarithms of both sides to help us find the exponent. It doesn’t matter which base we use in our logarithms. We can then use the laws of logarithms to help us. We know that log base 𝑎 of 𝑥 to the power of 𝑛 is equal to 𝑛 times log base 𝑎 of 𝑥. And therefore, we can write the right-hand side of this equation as 𝑛 minus one times log base 10 of four. We then divide through by log base 10 of four. And we can use our calculators to evaluate log base 10 of 262,144 divided by log base 10 of four, which gives us that 𝑛 minus one is equal to nine.
And so we have that 𝑛 is equal to 10. And this means that the order of the term must be 10. It’s the 10th term in the sequence, which has a value of 1,572,864. We could check our answer by filling in 𝑛 is equal to 10 into the general term. When we simplify the exponent, we get four to the power of nine, and four to the power of nine is 262,144. Multiplying this by six gives us the value of 1,572,864, which we were given in the question, which confirms our answer that this is the 10th term in the sequence.
