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