Find the sum of the infinite geometric sequence 49, minus 21, nine, et cetera.
First of all, when solving this problem, we just remind ourselves that a geometric sequence means a sequence that has a common ratio. So that means the ratio between consecutive terms is the same. And when we say an infinite series, we mean that our series will not have a last term, so there will be no last term.
In order for us to find the sum of this infinite geometric sequence, we need to use this formula, where 𝑆 is the sum, 𝑎 is the first term of our sequence, and 𝑟 is our common ratio. However, our common ratio does actually have some restrictions.
We don’t want our common ratio to be greater than minus one or less than one, or another way of thinking about is the absolute value or modulus of 𝑟 is less than one, and this is because if the value was either less than minus one or greater than one of 𝑟, then our sequence would actually keep getting larger and larger; each value would get larger and larger and larger. And therefore, we couldn’t actually reach a value for the sum.
So we’re now gonna work out the sum of our geometric sequence. So I’ve labelled the term numbers, our first, second, and third terms of our geometric sequence. What we’re gonna do first is we’re actually gonna find out what 𝑟 is at one, because obviously we need it to work out the sum, but also so that we can make sure it fulfils the parameters that we’ve already spoken about.
To find 𝑟, I’m gonna divide our second term by our first term, which gives us negative 21 over 49. Then we’re going to divide the numerator and the denominator by seven. Simplify it further, and this gives us negative three over seven, so we can say that our common ratio is negative three over seven.
Now we’ve found the common ratio; we can actually do a quick check. So we can look at negative three over seven, and yes it’s greater than negative one and it’s less than one. So therefore, we can use our formula. Okay so we’ve got 𝑟 is equal to negative three over seven. We just need to know what 𝑎 is.
Well 𝑎 is our first term, which in this sequence is 49. So we can now substitute these values into 𝑆 equals 𝑎 over one minus 𝑟, which gives us 𝑆 is equal to 49 over one minus negative three over seven, which gives us 49 over 10 over seven. Then we use our division law for fractions, which gives us 49 multiplied by seven over 10. Then we calculate that, and it gives us our sum of the infinite geometric sequence, which is 𝑆 is equal to 343 over 10.
So great, we’ve got our final answer, but we can now quickly recap what we’ve done. Well, first of all, we’re gonna find the common ratio, which is 𝑟, and that’s found by dividing a term by the previous term. Once we’ve got our common ratio, we just check that it’s between negative one, so it’s greater than negative one, and it’s less than one, just so that we know that we can in fact use the formula.
Then the other value we need for our formula is 𝑎, which is our first term, so we find that. And once we have those values, we substitute them into our formula: 𝑆 is equal to 𝑎 over one minus 𝑟. And then once we’ve done that, we know that we have the final answer, which is the sum of the infinite geometric sequence.