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