Question

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is n(n+1). If the nth term of the progression is divisible by 7, then the smallest possible value of n is:

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (B)

7

Answer 2

We know, nth term of an AP = Sum of n terms - Sum of (n-1) terms
Therefore, nth term = $n(n+1) - (n-1)(n) = n^2 + n - n^2 + n = 2n$
Given, nth term is divisible by 7. So, 2n must be divisible by 7.
For 2n to be divisible by 7, the smallest possible value of n is when $2n = 7\times2 = 14.$
Therefore, n = 7.