Question Video: Finding an Infinite Geometric Sequence given Its First Term and the Sum of All Its Terms Mathematics • 10th Grade

Video Transcript

Find the infinite geometric sequence given the first term is five over 57 and the sum of the terms is 20 over 171.

We have an infinite geometric sequence with first term five over 57 and sum of the terms 20 over 171. We know that the general form of a geometric sequence is 𝑎 one, 𝑎 one 𝑟, 𝑎 one 𝑟 squared, and so on, where 𝑎 one is the first term of the sequence, which in this case is five over 57, and 𝑟 is a common ratio. It’s what we multiply by to get the next term.

Another really useful formula that we have is the sum of a number of terms of an infinite geometric sequence, which is given by 𝑆 equals 𝑎 one over one minus 𝑟. And this is only valid when the absolute value of 𝑟 is less than one. In order to find what this geometric sequence is starting with the first term five over 57, we’re going to need to know what we need to multiply by to get each next term. And that’s where this formula is going to come in handy.

We know what 𝑆 is. It’s the sum of all the terms in the infinite geometric sequence. And we’re given in the question that this is 20 over 171. We also know 𝑎 one. It’s the first term of the sequence, and we’re told this is five over 57. But we don’t know what 𝑟 is. So let’s rearrange this for 𝑟.

We can start by multiplying both sides of the equation by one minus 𝑟. And then we can divide both sides by 20 over 171. So we have that one minus 𝑟 equals five over 57 over 20 over 171. We could think of this as five over 57 divided by 20 over 171. And from our rules of dividing fractions, we know that this is the same as five over 57 times 171 over 20, which, by multiplying the numerators and multiplying the denominators, we find to be 855 over 1140, which cancels down to three over four.

So we have that one minus 𝑟 is equal to three over four, which leads us to 𝑟 equals one over four. And notice how the absolute value of 𝑟 is less than one, as we said it needs to be to use the sum formula. Remember, we said that 𝑟 is the thing that we multiply by to find the next term in the sequence. So starting with the first term as five over 57, we find the next term by multiplying five over 57 by one over four. This gives us that the second term is five over 228. Then to find the third term, we multiply the second term by one over four. So we do five over 228 multiplied by one over four. And this gives us five over 912. And then, the sequence continues in the same way, by multiplying each term by one over four to find the next term.

So by using the formula for the sum of an infinite geometric sequence, we were able to determine what the common ratio is and therefore find the first few terms of this infinite geometric sequence.