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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.
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The geometric sequence is where each term is determined by multiplying by a non-zero constant, called a common ratio, by the previous term.
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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.
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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.
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And then we would take negative five sixteenths times the common ratio, we will get the next term, negative five sixty-fourths.
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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.
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So how do we find this common ratio 𝑥?
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And it’s the same for every single one.
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So we can just pick one of the sets and solve for 𝑥.
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Let’s go ahead and use the first one because it seems the simplest.
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So to solve for 𝑥, we need to divide both sides by negative five.
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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.
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So we’re multiplying by the reciprocal of negative five over one which is just where you flip it.
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So we’re multiplying by one over negative five.
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So when we multiply fractions, we multiply the numerators and we multiply the denominators.
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So on the top, we get negative five and on the bottom, we get negative 20.
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The negatives will cancel and then five twentieths reduces to one-fourth.
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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.
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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.
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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.
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And four and four, being multiplied on the bottom, gives you 16.
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And then negative five sixteenths times one-fourth is negative five sixty-fourths.
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And then lastly, we can find the last term by multiplying by that one-fourth.
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So the next term of the geometric sequence would be negative five two hundred and fifty-sixths.