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

Answer:

Let us consider:

AP 1 = 9, 7, 5, 3, ... and so on.

Here:

a = 9, d = ( -2 ), n = n

Let us consider:

AP 2 = 15, 12, 9, 6, ... and so on.

Here:

a = 15, d = ( -3 ), n = n

We know that:

$\huge\boxed{\sf{a_{n1} = a_{n2}}}$

By using this formula:

$\huge\boxed{\sf{a_{n} = a +(n-1)d}}$

We get:

$\implies$ $\sf{a_{n1} = 9 + (n - 1)( - 2)}$

$\implies$ $\sf{a_{n2} = 15 + (n - 1)( - 3)}$

Equating them:

$\implies$ 9 + ( n - 1 ) ( - 2 ) = 15 + ( n - 1 ) ( - 3 )

$\implies$ 9 + ( -2n + 2 ) = 15 + ( -3n + 3 )

$\implies$ 9 + 2 - 2n = 15 + 3 - 3n

$\implies$ 11 - 2n = 18 - 3n

Transposing all the n terms to one side we get:

$\implies$ 3n - 2n = 18 - 11

$\implies$ 1n = 7

$\implies$ n = 7

Therefore:

The value of n is 7.

Answer 2

Given :

A.P 1 : 9,7,5,..........

d = a2 - a1

= 7 - 9

= -2

A.P 2 : 15,12,9......

d = a2 - a1

= 12 - 25

= -3

an = a + (n-1)d

an1 = 9 + (n-1)(-2)

an2 = 15 + (n-1)(-3)

an1 = an2

9 + (n-1)(-2) = 15 + (n-1)(-3)

9 - 2n + 2 = 15 - 3n + 3

-2n + 3n = 18 - 11

n = 7

Therefore, the value of n is 7