Question

Let S_n denotes the sum of the first n terms of an AP and

S_2n = 3 S_n . Then find the ratio S_3n : S_n

Answer 1

S(n) denotes the sum of the first n terms. Its general formula is
an + n(n-1)d/2
where a is the first term and d is the common difference.

S(2n) = 3S(n)
2an + 2n(2n-1)d/2 = 3[an + n(n-1)d/2]
4an + 4n^2 d - 2nd = 6an + 3n^2 d - 3nd [multiplying both sides by 2]
4a + 4nd - 2d = 6a + 3nd - 3d [dividing both sides by n]
nd + d = 2a
d = 2a/(n+1)

S(3n):S(n)
= (3an + 3n(3n-1)d/2) : (an + n(n-1)d/2)
= (3an + 3n(3n-1)a/(n+1)) : (an + n(n-1)a/(n+1)) [substituting 2a/(n+1) for d]
= (3n(n+1) + 3n(3n-1)) : (n(n+1) + n(n-1)) [multiplying both sides by (n+1)/a]
= (3n^2 + 3n + 9n^2 - 3n) : (n^2 + n + n^2 - n)
= 12n^2 : 2n^2
= 6:1