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

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Solution

Given:
nth term of the A.P 9, 7, 5, .... is the same as the nth term of the A.P. 15, 12, 9, ....
Considering 9, 7, 5, ....
a = 9, d = (7 - 9) = - 2
nth term = 9 + (n - 1)(-2) [ $a_{n} = a + (n - 1) d$ ]
= 9 - 2n + 2
= 11 - 2n ..... (i)
Considering 15, 12, 9, ......
a = 15, d = (12 - 15) = - 3
nth term = 15 + (n + 1) (-3) [ $a_{n} = a + (n - 1) d$ ]
= 15 - 3n + 3
= 18 - 3n ..... (ii)
Equating (i) and (ii), we get:
11 - 2n = 18 - 3n
$\Rightarrow n = 7$
Thus, 7th terms of both the A.P.s are the same.

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