 Hello, and welcome to this screencast. In this video, we will find a general formula for the nth term of a sequence. Here we have the first four terms of a sequence. Four-sixth, negative five, thirty-sixth, six divided by two hundred and sixteen, and negative seven divided by twelve hundred and ninety-six. Our job is to identify a pattern in these terms, so that we can write a formula that we can use to determine any other term in this sequence. We notice that each of our terms is a fraction, so we start by writing a sub n with a line for our fraction. And first we're going to consider the numerators of our terms. We notice that the numerators include the consecutive numbers four, five, six, and seven, and we want to relate this pattern to the index, or the number, of each of our terms. So first, the numerator is four, when our index n is equal to one. The next numerator includes a five, when n is equal to two. Then we have a six in the numerator, when n is equal to three, and a seven in the numerator, when n is equal to four. In each term, the number in the numerator is three larger than the index for the term, so we will write n plus three in the numerator of our formula. But we know that that isn't the entire numerator, because we also notice that terms a sub two and a sub four have a negative sign in the numerator. We don't want all of the terms to be negative, but we do want the sign to alternate. And to get this result, we raise negative one to a power, because that will change the sign of our terms without changing their magnitudes. If we raise negative one to the nth power, then we would get a positive result whenever the exponent is even. But we want to get a negative result when n is even, so we will add one to the exponent. So when the index n is even, our exponent, n plus one will be odd, and we will get negative one raised to an odd power, which is negative one, and that's exactly what we want. The pattern in the denominators of our terms may be harder to see at first glance, but it turns out to be quite nice. Our first denominator is six, which we can write as six to the first power, and we notice that the index of our term is one. Our second denominator is thirty-six, which we easily recognize as six squared, and our index is n is equal to two. If we check, we see that this pattern will continue, so we notice that six to the third is equal to two hundred and sixteen, and six to the fourth is equal to twelve hundred and ninety-six. So in general, our denominator will be six to the nth power, where n is the index of our term. So we can use this formula that we developed, negative one to the n plus one power, times the quantity n plus three, divided by six to the nth, for integers n that are greater than or equal to one. And we can use this to find any term that we want in our sequence. To see how we can apply this formula that we just created, we will find the tenth term when n is equal to ten. So we write a sub ten is equal to, and for next we'll find our numerator. So we raise negative one to the power ten plus one, and we multiply by the quantity ten plus three, and then we divide by six to the tenth power. This looks kind of messy right now, so we can clean it up a little bit. When we simplify this, our numerator will be negative, because we have negative one to the eleventh power. And then we're going to multiply this by thirteen, so our numerator is simply negative thirteen. Six to the tenth is a pretty large number, but if we multiply it out, our result is sixty million, four hundred and sixty-six thousand, one hundred and seventy-six. And that's our final result for the tenth term in our sequence. And we could use this formula to find any other term in our sequence too. Thanks for watching.