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For what value of n, the $n^{th}$ term of the following two A.P's the same.

13, 19, 25, ... and 69, 68, 67, ...

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answer is D.

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Detailed Solution

Given two A.P.’s as 13, 19, 25, ... and 69, 68, 67, ...
We know, the formula of

n^{th}

term is,

T n = a + n − 1 d

.
The first term is a.
The common difference is d.
The number of terms are n.
Therefore, according to the first A.P.,

T n = 13 + n − 1 6 ⇒ T n = 13 + 6 n − 6 ⇒ T n = 6 n + 7

The

n^{th}

term of 1st A.P is

T n = 6 n + 7...... 1

.
Again, for the second A.P.,

T n = 69 + n − 1 − 1 ⇒ T n = 69 − n + 1 ⇒ T n = 70 − n

The nth term of A.P is

T n = 70 − n ...... (2)

Equating equations (1) and (2),

6 n + 7 = 70 − n ⇒ 6 n + n = 70 − 7 ⇒ 7 n = 63 ⇒ n = 9

The 9th term is a common term to both series 13,19,25,....... and 69,68,67,........
Therefore, option 4 is correct.

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