To solve this problem, we need to find the ratio of the sum of 11 terms of two arithmetic progressions (APs), given that the ratio of the sum to \(\( n \)\) terms of the two progressions is \(\( \frac{7n + 4}{6n - 5} \)\).
Here is the step-by-step solution:
1. Understand the Ratio Given for n Terms:
The ratio of the sum to \(\( n \)\) terms of the two arithmetic progressions is given by:
\(\[
\text{Ratio} = \frac{7n + 4}{6n - 5}
\]\)
2. Substitute \(\( n = 11 \)\):
We need to find this ratio specifically for \(\( n = 11 \)\):
\(\[
\text{Ratio for 11 terms} = \frac{7 \times 11 + 4}{6 \times 11 - 5}
\]\)
3. Calculate the Numerator:
\(\[
7 \times 11 + 4 = 77 + 4 = 81
\]\)
4. Calculate the Denominator:
\(\[
6 \times 11 - 5 = 66 - 5 = 61
\]\)
5. Form the Ratio:
\(\[
\frac{81}{61}
\]\)
Hence, the required ratio of the sum of 11 terms of the two arithmetic progressions is \(\( \frac{81}{61} \)\).
So, the correct option is:
\(\[
\boxed{81/61}
\]\)
The correct answer is Option C.
