Let S k denote the sum of the first k terms of an A.P. If the sum of its first 3n terms is twice the sum of the next n terms, and the sums in Statement 1 are nonzero, assess: Statement 1: The ratio

Question

Let S k ​ denote the sum of the first k terms of an A.P. If the sum of its first 3n terms is twice the sum of the next n terms, and the sums in Statement 1 are nonzero, assess: Statement 1: The ratio of the sum of the first 2n terms to the sum of the next 2n terms is 7:11. Statement 2: S n ​ , S 2 n ​ , S 3 n ​ are in A.P.

TestHub · Accepted Answer

Write S k ​ = k [ 2 a + ( k − 1 ) d ] /2 and put B = nd , C = 2 a − d . Then S jn ​ = jn ( C + jB ) /2 . The condition 3 S 3 n ​ = 2 S 4 n ​ gives C=5B. Thus S n ​ : S 2 n ​ : S 3 n ​ = 6 : 14 : 24 , not A.P.; S 2 n ​ : ( S 4 n ​ − S 2 n ​ ) = 14 : 22 = 7 : 11 .

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