Question

If the sum of first n terms of two A.P.' s are in the ratio 3n + 8 : 7n + 15, then the ratio of their 12th term is

• A. 7:16
• B. 16:7
• C. 23:55
• D. 55:23

Answer 1

Answer: (A)

7:16

Answer 2

The ratio of the sum of the first $n$ terms of two APs is given by $\frac{S_n}{S'_n} = \frac{2a_1 + (n-1)d_1}{2a'_1 + (n-1)d'_1}$. To find the ratio of the $k^{th}$ terms, $\frac{T_k}{T'_k} = \frac{a_1 + (k-1)d_1}{a'_1 + (k-1)d'_1}$, we equate $\frac{n-1}{2} = k-1$, which implies $n = 2k-1$. For the $12^{th}$ term ($k=12$), we substitute $n = 2(12)-1 = 23$ into the given ratio of sums: $\frac{3(23) + 8}{7(23) + 15} = \frac{69 + 8}{161 + 15} = \frac{77}{176}$. Simplifying the ratio by dividing both numerator and denominator by 11, we get $\frac{7}{16}$.