Find a formula for the general term an of each of the...

Find a formula for the general term $a_n$ of each of the following sequences. 12. $\{1, 0, -1, 0, 1, 0, -1, 0, ...\}$ (Hint: Find where $\sin x$ takes these values) 13. $\{1, -1/3, 1/5, -1/7, ...\}$

Transcript

00:01 In this problem, we have been asked to find a formula for the general term a -n for each of the given sequences.

00:08 Now, the full sequence given is 1 -0 -1 -10 -1 -10 -1 -0 and so on.

00:21 So this is our first sequence.

00:24 Now, let us rewrite the terms of the sequence.

00:27 Now, first of all, we can write 1 as sine of pi by 2.

00:32 We can write 0 as sign of pi.

00:36 Now, negative 1 is sign of 3 pi by 2, and 0 can be written as sign 2 pi.

00:45 Similarly, the next 1 can be written as sign of 2 pi plus pi by 2, so that will be 5 pi by 2.

00:57 And the next one which is 0 we can write that as sign of 3 pi and for minus 1 we can write sign 7 pi 2 for the 0 we can write sign of 8 sign 4 pi and so on so we can also rewrite these so let us rewrite pi as 2 times pi by 2 let us rewrite 2x 2 5 5 5.

01:25 Let us rewrite 2 pi as 4 times pi by 2.

01:36 Let us rewrite 3 pi as 6 times pi by 2.

01:41 Let us rewrite 4 pi as 8 times pi by 2.

01:48 So, and for pi by 2, let's write that as 1 times pi by 2.

01:53 So we can see that we have a pattern.

01:55 Sign of 1 times pi by 2, sign of 2 times pi by 2, sign of 3 times pi by 2, 4 times pi by 2 and so on.

02:03 So we can use this pattern to write the general term of the sequence, a .n is equal to sine of n pi by 2.

02:13 So you can see that the first term is obtained by substituting n equals to 1.

02:16 We get this.

02:17 If we substitute n equals to 2, 3, 4, 5, 6, 7, 8.

02:23 So if we just keep on substituting it, we will get our required terms.

02:26 So our required expression for a .n.