Beginning with n = 1, write the first four terms of the...

Beginning with n = 1, write the first four terms of the sequences whose nth terms, or general terms, are given by: a) an = 3+(-1)^{n+1} b) an = frac{(n+1)!}{2^n} A sequence defined recursively has the formula a_1 = -2 and a_n = -3a_{n-1}+4 for n ge 2. Compute the first four terms of the sequence. Find the value of the sum sum_{k=1}^5 (k-2)(k-3) Express in summation notation beginning with 1 as the lower limit of summation: 4 - frac{5}{3} + frac{6}{9} - frac{7}{27} + frac{8}{81} - frac{9}{243}

Transcript

00:01 There's a lot of work to do in this problem.

00:02 So i'm just going to jump into part a where the first term is plugging in 1 in for n.

00:12 And 1 plus 1 is equal to 2.

00:15 In the moment that you square a negative number, you're going to end up with a positive one.

00:22 So the first term is 4.

00:24 So may i'll just write it like i'll write in order.

00:27 So it'll be 4 comma.

00:29 So what changes is when you go ahead and move one up to two and you add one to it.

00:37 Well, now you're doing three plus negative one cubed, which three negatives makes it negative again.

00:47 So it's going to be two.

00:48 And if i continue doing this and change this to the fourth hour because it's the third term, you're going to get four.

00:55 And then two again because we change this to four plus one and be five and back to two.

01:02 So there's the first four terms for a.

01:05 Moving on to b, where you have 1 plus 1 factorial, because i'm substituting 1 in for n, and then 2 to the first power.

01:19 Well, you end up with 2 factorial over 2, so 2 divided by 2 is 1.

01:26 So again, i'm going to scale and list it out.

01:28 So then moving on to the second term, it would be 2 plus 1 factorial over 2 to the second power.

01:35 So that's 3 factorial, which i'm assuming you know is 3 times 2 times 1.

01:42 And that would reduce 6 .4 reduces the 3 halves.

01:47 So moving on to the next one, i'm just going to kind of jump ahead that it will be 4 factorial over 2 cubed.

01:55 So 4 times 3 times 2 times 1 divided by 8.

02:00 Well, 4 times 2 equals 8.

02:02 So you're just left with 3 times 1, which would be 3.

02:04 And then just one more it would be five factorial over two to the fourth and i've done five factorial enough to know that that's 120 248 16 and i don't know if that divides 120 over 16 reduces to 15 halves so that's that one so moving on to number two yeah this is going to be a long video sorry so they tell you the first term is negative two and they want you to compute the first four terms of the secret.

02:40 Okay, so we already know that the first term is negative two.

02:43 I need to figure out the next couple.

02:46 And what it means to be recursive is you need the previous number, i'll do this in blue, in order to figure out the next one.

02:54 So the pattern is negative three times the previous term and then plus four.

03:05 So you're looking at six plus four would be ten.

03:10 And so the pattern stays the same, except what you have to do is take that 10 and plug it in here...

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