Everyone, welcome back. In this example, we're going to do something very similar where you have the first five numbers of a sequence: -2, 4, -6, 8, -10, and so on and so forth. In this problem, we're asked to write the general formula for the nth term, and we're going to use this to calculate the 18th term. These formulas are really useful for calculating really high terms because rather than having to follow the pattern for 18 terms, we're going to be able to figure out a formula for an. Let's go ahead and get started here.
Let's look at the pattern of numbers. We don't see any fractions. From -2 to 4, there's an increase of 2. I'm just looking at the number, not the sign. From 4 to 6, there is also an increase of 2, from 6 to 8, an increase of 2, and from 8 to 10, another increase of 2. We've got the same number increasing each time, but we also have an alternating sign pattern. When you have alternating signs, that always means that there's going to be -1 to the nth power. If the first term is negative, then it's just going to be -n, not n + 1.
We need to figure out how to increase these numbers by 2 each time. If we multiply this sequence by 2n, it will account for the fact that each term, as the index rises, will increase by 2. For a1, this would be -1 to the first power times 2 times 1, which is -2, matching the first term. For a2, it's -1 to the second power times 2 times 2, equating to 4, matching the second term. Checking a3, -1 to the third power times 2 times 3 results in -6, exactly as our third term. The general formula for the sequence is -1 to the nth power times 2n.
To find the 18th term, we simply plug in 18 for n in our formula. The 18th term is -1 raised to the 18th power times 2 times 18, which results in 36. Thanks for watching, hopefully this made sense, and I'll see you in the next one.