Find the next term of the geometric sequence negative five, negative five-fourths, negative five sixteenths, negative five sixty-fourths, and then we will be finding that last term.
The geometric sequence is where each term is determined by multiplying by a non-zero constant, called a common ratio, by the previous term. So if we begin with negative five, we would multiply by this common ratio we don’t know, as 𝑥, and we would get the next term, negative five-fourths. And then we would take negative five-fourths multiplied by the common ratio 𝑥 which we will find, and we will get the next term, negative five sixteenths. And then we would take negative five sixteenths times the common ratio, we will get the next term, negative five sixty-fourths. And then lastly, we would take negative five sixty-fourths times the common ratio to get our last term, what we’re trying to find.
So how do we find this common ratio 𝑥? And it’s the same for every single one. So we can just pick one of the sets and solve for 𝑥. Let’s go ahead and use the first one because it seems the simplest. So to solve for 𝑥, we need to divide both sides by negative five. When you’re taking fractions and dividing, because really the negative five is a negative five over one, instead of dividing by a fraction, you multiply by its reciprocal. So we’re multiplying by the reciprocal of negative five over one which is just where you flip it. So we’re multiplying by one over negative five.
So when we multiply fractions, we multiply the numerators and we multiply the denominators. So on the top, we get negative five and on the bottom, we get negative 20. The negatives will cancel and then five twentieths reduces to one-fourth. So one fourth is that common ratio that we’re going to be multiplying by to every single term so we can get the next one.
Just to double-check, if we would plug in one-fourth in for 𝑥 back into each of these to find the next term, it does indeed work. Negative five times one-fourth is negative five-fourths, negative five-fourths times one-fourth is negative five sixteenths because the top multiplies be it negative five. And four and four, being multiplied on the bottom, gives you 16. And then negative five sixteenths times one-fourth is negative five sixty-fourths. And then lastly, we can find the last term by multiplying by that one-fourth.
So the next term of the geometric sequence would be negative five two hundred and fifty-sixths.