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

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 = n^2/n!

Accepted Answer

Hello everyone in this video. We're going to be looking at this practice problem where we want to write the sequences first four terms. Where the sequence A is equal to n plus one squared divided by N plus two factorial. So in this case because we want the first four terms we want from where N equals one through and equals four. And we're going to be plugging in and equals one through and equals four into the expression that they give us which is nine plus one squared divided by N plus two factorial. So I'm going to start by evaluating and equals one. And if I plug in one for the value of N to the expression I get one plus one squared divided by one plus two factorial. We'll start simplifying this. My numerator becomes two squared divided by three factorial and two squared is equal to four. And recall that factorial are the product of an integer and all the integers below it. So if I think about the integers below three I have to multiply by two and by one. So this becomes four divided by three times two is six. And I can simplify this 4. 6 is the same as two thirds. So when N is equal to one, get 2/3. So 2/3 is my first term and I can continue solving when n is equal to two more for my second term my expression is two plus one squared divided by two plus two factorial. If I start simplifying this two plus one is three. So I have three squared in the numerator and two plus two is four. So I have four factorial in my denominator three squared is nine and four factorial is the same as four Times three times 2 times one. And that is equal to nine divided by four times three is 12 times two is 24. So I have nine divided by 24 and I can simplify this fraction Because they both have factors of three so I can write it as 3/8. My third term is an equals three. So when I plug three into the value of N my expression is three plus one squared divided by three plus two factorial. I start simplifying three plus one is four. So I have four squared in the numerator and three plus two is five factorial and the denominator four squared is 16 divided by five factorial. So five factorial I can write as five times four times three times two times one. But because we're getting to these longer digits I'm going to simplify this and write it as five times four factorial because I've already solved for four factorial appear I got 24. So it just makes simplifying quicker because now I know this is 16 divided by five times 24. So this becomes 16 divided by five times 24 which is 1 20. And once again I can reduce this because both 16 and 1 20 have a factor of eight. So I can reduce this to to over 15. So now I have my first term, my second term, my third term. And now I just need my fourth term Burton is equal to four. So when I have an is equal to four my expression becomes four plus one squared divided by four plus two factorial. If I simplify that I have five squared in my numerator and six factorial in my denominator five squared is 25 and I'm going to do the same as before. But now I have six times five factorial and I solved the value of five factorial in my previous term. So this becomes 25 divided by six times 1 20. And I simplify my denominator. I get 25 divided by 7:20 And both 25 and 720 have a factor of five. So I can simplify this fraction to five divided by 144. And now this is my fourth term. So let me pull all my terms out from this work. So my first term is 2/3. My second term is 3/8. My third term is to 15th and my fourth term is 51 44th 144th. So this becomes my final answer. And notice how I reduced the amount of work I did by relying on the previous factorial that I had already solved and looking at our answer. You can see that this aligns with answer choice B So flee this video helped and see you next time.

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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 = n^2/n!