A man sent a message to two of his friends, and each of them then sent the same message to another two friends and so on. Find the number of people who got the message the sixth time it was sent given that each person received the message only once.
Let’s have a look at what’s happening here. The very first time two people received the message. Then each person sent two of their friends the same message. That means two times two or four people received the message the second time. Then each of those four people sent the message to two of their friends, meaning eight people received the message the third time. We could continue this process, multiplying each value by two to work out the number of people who receive the message each time. In fact, we’re looking to find the number of people who got the message the sixth time. So we could just carry this on until we reach the sixth term.
However, if we look very carefully, we see we have a special type of sequence. We call this sequence a geometric sequence. It’s one where each term is found by multiplying the previous term by some fixed value. And we call that fixed value the common ratio. In this case, if we define 𝑎 to be equal to the first term, that’s two, and 𝑟 to be equal to the common ratio, that’s also two, we recall that we can use the 𝑛th term for a geometric sequence to find any term in the sequence. It’s 𝑎 sub 𝑛 equals 𝑎 times 𝑟 to the power of 𝑛 minus one.
We want to find the number of people who got the message the sixth time, so we’re going to let 𝑛 be equal to six. The sixth term is then 𝑎 sub six. And it’s found by multiplying the first term, two, by the common ratio to the power of 𝑛 minus one. So that’s two to the power of six minus one. That’s two times two to the fifth power. Then we can use the laws of exponents. And we see that two to the power of one times two to the power of five is two to the power of one plus five or two to the sixth power. And of course, two to the sixth power is 64. And so we can see that the number of people who got the message the sixth time must be 64.