WEBVTT
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Find the fourth term in the sequence whose first three terms are one, negative one-half, and one-third.
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We’re going to begin by rewriting the numbers in our sequence just a little bit.
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We can write one as a fraction.
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We’re going to write it as one over one.
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And then, we’re going to consider what’s happening to the numerators and the denominators in our sequences individually.
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We’ll begin with the denominator because that’s a little bit easier.
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The first three denominators are one, two, and three.
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These have a common difference of one.
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So, we can see that the denominator of our fourth term must simply be equal to four.
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But what about the numerators?
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Well, negative one-half we can consider as being the same as negative one over two.
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And so, we could say that the numerators are one, negative one, and one.
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But what’s happening here?
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Well, these terms are oscillating.
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That is, they’re increasing and decreasing.
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To get from the first term to the second term, we take away two.
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And then, to get from the second to the third, we add two.
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It follows to get from the third term to the fourth term, we’d subtract two.
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And that gives us a fourth numerator of negative one.
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And we can put these together and say that this means the fourth term in our sequence must be negative one over four or negative one-quarter.
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The fourth term is negative one-quarter.