(10) 5. In each of the following, determine if it is...

(10) 5. In each of the following, determine if it is arithmetic, geometric, or neither, list the next three terms and then write the $n^{th}$ term. a) 8, 11, 14, 17, 20, ... 23, 26, 29 arithmetic b) 5, 15, 45, 135, 405, ... geometric

Transcript

00:01 Hi here for the given question we are given a sequence which is 1 ,3 ,9 ,19 ,33 ,51 and so on.

00:10 So here by taking the difference value we have sequence as 2 ,6 ,10 ,14 ,18 and so on.

00:19 So here the difference sequence is 4 ,4 ,4 and so on.

00:23 So here now we know that sequence of squared numbers can be written as 1 ,4 ,9 ,16 ,25 and so on.

00:36 So here now further we can observe that here the consecutive difference is 3 ,5 ,7 ,9 and so on.

00:51 So here now using this value we can say that a n will be equal to n square plus b.

00:57 So here if we take n is equal to 0 we have a0 is equal to 0 square plus b which is equal to 1.

01:05 So here this is the value of our first term.

01:08 Now here if we take the value as a1 n is equal to 1 then we have 1 square plus b which is equal to 3 which implies b is equal to 2 and so on.

01:24 So here in our case we can say that our required sequence a n is equal to n square plus 1.

01:32 So here this is the required first solution.

01:36 Now further for the next part we are given that here we have sequence which is 2 ,3 ,10 ,29 ,66 ,127 and so on.

01:48 So here the difference is 1 ,7 ,19 ,37 ,61 and so on.

01:55 Again difference between them is 6 ,12 ,18 ,24 and so on.

02:02 Now here if we consider a cubic sequence then here we have 1 ,8 ,27 ,64 ,125 and so on.

02:15 Now here difference between this sequence is 12 ,18 ,24 and so on.

02:21 So here from this sequence we can say that here in our case we have a n is equal to n cube plus c.

02:29 So here in our case now if we take n is equal to 1 we have a1 is equal to 1 cube plus c is equal to 2 which implies c equals to 1.

02:40 So here in our case we can say that a n is equal to n cube plus 1 is our required solution.

02:46 Now similarly we need to check for the third sequence...

Question

(10) 5. In each of the following, determine if it is arithmetic, geometric, or neither, list the next three terms and then write the $n^{th}$ term. a) 8, 11, 14, 17, 20, ... 23, 26, 29 arithmetic b) 5, 15, 45, 135, 405, ... geometric

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