Question

If the sums of n terms of two arithmetic series are in the ratio 2n + 3 : 6n + 5, then the ratio between their 13^th terms is

• (A) 53 : 155
• (B) 27 : 87
• (C) 29 : 83
• (D) 31 : 89

Answer 1

The Correct Option is A

Explanation:
Given that the sum of $n$ terms of two arithmetic series is in the ratio $2n+3:6n+5$$$\Rightarrow \frac{{\left(S_n\right)}_1}{{\left(S_n\right)}_2}=\frac{2n+3}{6n+5}$$where $S_n$ be the sum of $n$ terms of an arithmetic series. We know that$$S_n=\frac{n}{2}[2a+(n-1)d]$$From Eq. (i), we get$$\begin{array}{l}\frac{{\left(S_n\right)}_1}{{\left(S_n\right)}_2}=\frac{\frac{n}{2}[2a_1+(n-1)d_1]}{\frac{n}{2}[2a_2+(n-1)d_2]}=\frac{2n+3}{6n+5}\\ \Rightarrow \frac{2a_1+(n-1)d_1}{2a_2+(n-1)d_2}=\frac{2n+3}{6n+5}\\ \therefore \frac{2a_1+(25-1)d_1}{2a_2+(25-1)d_2}=\frac{53}{155}\\ \Rightarrow \frac{a_1+12d_1}{a_2+12d_2}=\frac{53}{155}\\ \Rightarrow \frac{{\left(T_{13}\right)}_1}{{\left(T_{13}\right)}_2}=\frac{53}{155}\end{array}$$