What kind of sequence is the following: one, one tenth, one one hundredth, one one thousandth, and so on?
There are two different kinds of sequences. It could be arithmetic or geometric. In an arithmetic sequence, each term is determined by adding a constant value to the previous term. We call that the common difference. So we can check if there’s a common difference. In a geometric sequence, each term is determined by multiplying a non-zero constant by the previous term. And we call that the common ratio. So let’s go ahead and check for a common difference and then also check for a common ratio. This will help determine what kind of sequence it is.
So first, let’s look at arithmetic. Essentially, we’re adding something to each term to get the next one. So what do we keep adding by? What we can check by, taking a term and subtracting the previous term from it. So we can take one tenth minus one. And one is the same as 10 over 10. So one minus 10 on the numerator would be negative nine, and then we keep our denominator. So we’ve negative nine tenths.
So now let’s check another set. So if we would subtract one tenth from one one hundredths, we should also get negative nine tenths, if this is arithmetic. One tenth is the same as ten one hundredths. And then we subtract the numerators, we get negative nine. However, the denominator is 100. And negative nine tenths is not the same thing as negative nine one hundredths. So this is not arithmetic.
Let’s check geometric. So now we’re doing the exact same thing except instead of subtracting, we’re making sure that they would divide to be the same thing. Because if we’re multiplying each one to find what we multiplied by, you have to divide. One tenth divided by a one is just one tenth because dividing by one doesn’t do anything. Now when we take one one hundredth divided by one tenth, we actually need to multiply it by 10 over one. Because when you divide fractions, you multiply by the reciprocal. And we get ten one hundredths. So they’re both one tenth.
So this sequence would be geometric only.