Question

If the sum of \( n \) terms of two A.P.s are in the ratio \( (3n + 8) : (7n + 15) \), then the ratio of their \( 12^{\text{th}} \) terms is:

• A. $8:7$
• B. $7:16$
• C. $44:99$
• D. $1:2$

Hint:

To get ratio of $m^{th}$ terms from ratio of $S_{n}$, always use $n = 2m - 1$.

Answer 1

The Correct Option is B

Step 1: Concept
Ratio of $n^{th}$ terms can be found from the ratio of sum of $n$ terms by replacing $n$ with $2n-1$.
Step 2: Analysis
To find the ratio of the $12^{th}$ terms ($T_{12}$), substitute $n = 2(12) - 1 = 23$ into the sum ratio.
Ratio $= \frac{3(23) + 8}{7(23) + 15} = \frac{69 + 8}{161 + 15}$.
Step 3: Conclusion
Ratio $= \frac{77}{176} = \frac{7 \times 11}{16 \times 11} = 7:16$.
Final Answer: (B)