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=2(n+1)!

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to write a sequences first four terms and in this case the sequence is a and it's equal to three times two N minus one factorial. So because we want to write the first four terms we're going to write for N equals one through n equals four. So let's go ahead and start evaluating those. So when N is equal to one I'm gonna be plugging in the values of N into the expression three times two N minus one factorial. So if I plug in one for N I get three times two times one minus one factorial. If I start simplifying this, I'm left with three times two times 1 is 2 -1 is one. So one factorial Is equal to one. So this becomes three times 1. So when N is equal to one or my first term I'm left with three. If I continue that process with N is equal to two, I plug in two for N and I get three times two times two minus one factorial. If I simplify this I get three times times two is 4 -1 factorial and if I continue I get three times four minus one is three factorial. So now I have three times three factorial and three factorial is three times all the integers below three which in this case we have two and one below three. So now I have three times two is six times one is still six. So now I have three times six, so when N is equal to two I get 18. Now when N is equal to three, we got three times two times 3 -1 factorial. If I start simplifying I get three Times two times 3 is 6 -1 is five factorial. So I get three times five factorial which is five times all the integers below it. So 432 and one and I can start multiplying bees. So I get three times five times four is 20 and three times two is six times we got the same six. So this becomes three times 1 20 Which evaluates to 360 and finally we have and is equal to four. So when I plug that in To my expression I get three times 2 times 4 - factorial. I start simplifying, I got three times two times four is eight minus one is seven factorial. So this becomes three times seven factorial, which is seven times six times five times four times three times two and times one. So if I start doing those multiplication zai get three times seven times six is 42 times five times four is 20 times three times two is six. So this is equal to three times 42 times 1, 20. And This evaluates to three times 5040. And when we multiply 5040 times three We got 15,120. So now we have solved the first four terms for the general sequence where the first term is three, the second term is 18, the third term is 360. The 4th term is 15,120. So this becomes our final answer for this problem. And you can see that this aligns with answer choice D. So hopefully 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=2(n+1)!

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