Question

what is the next term: 2,11,36,85,206,-----?

Answer 1

Answer:

375

Step-by-step explanation:

Here in this series we can se that the differences between each term can be written as

difference between

2 and 11 is 9

11 and 36 is 25

36 and 85 is 49

85 and 206 is 121

in all these differences one thing is common that they are square of some numbers

like 9 is of 3, 25 is of 5, 49 is of 7 and 121 is of 11

and when we look at the square roots of these numbers we can find a series of prime numbers

3,5,7,9,11 and after that 13

and the square of 13 is 169 so we add 169 to 206 because it is a increasing series

so when we add 169 and 206 we get 375 that's the answer.

Answer 2

Given

Ap terms :2,11,36,85,206

To Find

Next term after 206??

$\sf\huge\bold{\underline{\underline{{Solution}}}}$

Now, a is the first terms which is 2

$a_{2}$ is the second term which is 11

D is the common difference between =11-2=9

Now,

=>11-2=9

=>36-11=25

=>85-36=49

=>206-85=121

We observe that the difference of two APs first and second term is 9,third and second term is 25,fourth and third term is 49 & fifth and fourth term is 121.

9 is the square of 3

25 is the square of 5

49 is the square of 7

121 is the square of 11

The gap between 3&5 is 2 ;7&5 is 2 but between 11 &7 is 4 .

From above all we observe that the difference is of prime numbers so,the next term will be square of 13.e.g, =>13²=169

169-206=-37 [AP can't be negative so, we add 169 to 206 because Given AP is in increasing order]

=>169+206=375

Therefore,the next term will be 375

Check

375-206=169 [(13)²=169]

Before this we obtained 121 [square of 11]

We are right according to our observations