# For what value of n, the nth terms of the arithmetic progressions 63, 65, 67, ... and 3, 10, 17, ... are equal?

Let a1 and d1 be the first term and common difference of the AP 63, 65, 67, 69,….

Let a2 and d2 be the first term and common difference of the AP 3, 10, 17,….

a1 = 63, d1 = 2

a2 = 3, d2 = 7

Let an be the nth term of the first AP and bn be the nth term of the second AP.

So, an = a1 + (n - 1)d1

an = 63 + (n - 1)2

an = 61 + 2n

and, bn = a2 + (n - 1)d2

bn = 3 + (n - 1)7

bn = - 4 + 7n

Since for nth terms of both the AP’s to be same, an = bn

61 + 2n = - 4 + 7n

61 + 4= 7n - 2n

65 = 5n

n = 13

Therefore, 13th term of both the AP’s will be same.

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