For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equal - Mathematics

For what value of n, are the nth terms of two APs 63, 65, 67, and 3, 10, 17, … equal?

Solution 1

Consider the A.P 63, 65, 67, …

a = 63

d = a2 − a1 = 65 − 63 = 2

nth term of this A.P. = an = a + (n − 1) d

an= 63 + (n − 1) 2 = 63 + 2n − 2

an = 61 + 2n (1)

3, 10, 17, …

a = 3

d = a2 − a1 = 10 − 3 = 7

nth term of this A.P. = 3 + (n − 1) 7

an = 3 + 7n − 7

an = 7n − 4 (2)

It is given that, nth term of these A.P.s are equal to each other.

Equating both these equations, we obtain

61 + 2n = 7n − 4

61 + 4 = 5n

5n = 65

n = 13

Therefore, 13th terms of both these A.P.s are equal to each other.

Solution 2

Consider the A.P 63, 65, 67,...

Here, First term (a) = 63 and common difference (d) = 2

an = a + (n − 1)d

an = 63 + (n − 1)2

an = 61 + 2n .....(1)

Consider another A.P 3, 10, 17,...

Here, First term (a') = 3 and common difference (d') = 7

a_n^'= a' + (n − 1)d'

a_n^' = 3 + (n − 1)7

a_n^' = 7n − 4 .....(2)

According to question,

61 + 2n = 7n − 4

⇒ 65 = 5n

⇒ n = 13

Concept: nth Term of an AP
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NCERT Class 10 Maths
Chapter 5 Arithmetic Progressions
Exercise 5.2 | Q 15 | Page 106