In Exercises 19–22, the general term of a sequence is given an...

Question

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=−2(n−1)!

Accepted Answer

Hello, today we are going to be writing the first four terms of the general term of the given sequence. The general term is given to us as a sub N is equal to negative 13 multiplied by N plus two factorial. So how do we write out the first four terms of this general term? In order to write out the first four terms, we are going to take values of N starting from one and going to four and plugging it in to the general term to start this process, we'll begin with N is equal to one. If we take this value of N and plug it into the generic term, we will get a sub one is equal to negative 13 multiplied by one plus two factorial. What we will have to do now is we will have to algebraically simplify the given term. So one plus two will give us three. So we will be left with negative 13 multiplied by three factorial. Next, what we need to do is we need to simplify the factorial. Now, in case you were having trouble simplifying a factorial, let's say for example, we are given the number four factorial four factorial is the product of four. And all of the terms that come before it. So in order to simplify four factorial, we can rewrite the factorial to be four multiplied by three, multiplied by two, multiplied by one. And we're going to expand the factorial in this way in order to simplify it. So let's go ahead and first write the coefficient of negative 13. And once we expand three factorial, we can rewrite the factorial to be three multiplied by two, multiplied by one, three, multiplied by two, multiplied by one will give us the value of six. So we are left with negative 13 multiplied by six. And the product of these two numbers will give us negative 78. This is going to be the first term of the generic term A sub N. Now what we are going to do is we're going to repeat this process but with the value N is equal to two. If we plug in the value N is equal to two, we will end up with a sub two is equal to negative 13, multiplied by two plus two, factorial two plus two will give us the value of four. So we will now have negative 13 multiplied by four factorial just like before, we will simplify the factorial by expanding it. And we can rewrite four factorial to be four multiplied by three, multiplied by two, multiplied by one. The product of four multiplied by three, multiplied by two, multiplied by one will give us the value of 24. So we will be left with negative 13 multiplied by 24. And the product of these two numbers will give us negative 312. This will be the second term of the generic term. A sub N. Once again, we will repeat this process but with N is equal to three. When N is equal to three, we get a sub three is equal to negative 13, multiplied by three plus two. Factorial three plus two will give us the value of five. So we have negative 13 multiplied by five factorial. We will then expand five factorial by rewriting it as a product. So we will have negative 13 multiplied by five, multiplied by 432 and one. The product of five multiplied by four, multiplied by three, multiplied by two, multiplied by one will give us the value of 120. So we will be left with negative 13 multiplied by 120. And the product of these two numbers will give us negative 1560. We will then get our final term by plugging in the value and is equal to four. When N is equal to four, we will have a sub four is equal to negative 13, multiplied by four plus two. Factorial. This will simplify to leave us with negative 13 multiplied by six factorial. We will then expand six factorial as six multiplied by five, multiplied by 432 and one. The product of all the numbers in the group will give us 720. So we will be left with negative 13 multiplied by 720. And the product of these two numbers is going to be negative 9360 which will be the final term. So just to recap in order to get the first four terms of the generic term, A sub N, we plugged in values of N from 1 to 4. When doing this, our first term was negative 78. Our second term is negative 312. Our third term is negative 1560. And our final term is negative 9360. And this is going to be the first four terms of the generic term A sub N. And with that being said, the answer to this problem is going to be B So I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=−2(n−1)!

Key Concepts

Factorial

Sequence

General Term of a Sequence

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