Question

If the nth term of the A.P. 9, 7, 5, ... is same as the nth term of the A.P. 15, 12, 9, ... find n.

Answer 1

Here, we are given two A.P. sequences whose n^th terms are equal. We need to find n. 

So let us first find the n^th term for both the A.P.

First A.P. is 9, 7, 5 …

Here

First term (a) = 9

Common difference of the A.P. (d) = 7 - 9

= -2

Now as we know

`a_n = a + (n - 1)d`

So for nth term

`a_n = a + (n -1)d`

So for nth term

`a_n = 9 + (n -1)(-2)`

= 9 - 2n + 2

= 11 - 2n  .......(1)

Second A.P. is 15, 12, 9 …

Here,

First term (a) = 15

Common difference of the A.P. (d) = 12 - 15

= -3

Now as we know

`a_n = a + (n - 1)d`

So for nth term

`a_n = 15 + (n -1)(-3)`

= 15 - 3n + 3

= 18 - 3n .....(2)

Now, we are given that the n^th terms for both the A.P. sequences are equal, we equate (1) and (2),

11 - 2n = 18 - 3n

3n - 2n = 18 - 11

n = 7

Therefore n = 7