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  • If Sndenotes the sum of first n terms of A.P and S2n=3Sn then S3n/ Sn is equal to ?

If Sndenotes the sum of first n terms of A.P and S2n=3Sn then S3n/ Sn is equal to ?

Answers (1)

best_answer

Let the first term be a and common difference be d.

Then, S2n = 3Sn 

Then

   \begin{array}{l} \Rightarrow \frac{2 n}{2}[2 a+(2 n-1) d] =\frac{3 n}{2}[2 a+(n-1) d] \\ \Rightarrow 2[2 a+(2 n-1) d] =3[2 a+(n-1) d] \\ \Rightarrow 4 a+(4 n-2) d =6 a+(3 n-3) d \\ \Rightarrow 2 a=(n+1) d \end{array}

Now

\begin{aligned} \frac{S_{3 n}}{S_{n}} &=\frac{\frac{3 n}{2}[2 a+(3 n-1) d]}{\frac{n}{2}[2 a+(n-1) d]} \\ &=\frac{3[2 a+(3 n-1) d]}{ [2 a+(n-1) d]} \\ &=\frac{3[(n+1) d+(3 n-1) d]}{[(n+1) d+(n-1) d]} \\ &=\frac{3[4 n d]}{2 n d}=6 \end{aligned}

Posted by

avinash.dongre

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