If $\frac{1}{m} + \frac{2}{m} + \frac{2^2}{m^2} + \ldots + \frac{2^n}{m^n} = \frac{s}{7}$, then the value of $m + n$ is
Step-by-Step Solution
Key Concept: Binomial coefficient identities and sums of specific binomial coefficients can be solved by setting up equations and testing integer solutions.
We have $\frac{1}{12}(\frac{10}{m} + \frac{10}{n} + \frac{10}{m})\cdot \frac{n}{2} = \frac{1}{12}(^nC_0 + ^nC_1 + ^nC_2) = \frac{n}{2}$. Expanding: $\frac{1}{12}(^nC_0 + ^nC_1 + ^nC_2 + ^nC_3 + ^nC_4 + ^nC_5) = \frac{n(n-1)}{2}$. Continuing with all terms: $\frac{1}{12}(^nC_0 + ^nC_1 + ^nC_2 + ^nC_3 + ^nC_4 + ^nC_5 + ^nC_6) = \frac{n(n-1)}{2}$. Testing $n = 10$ and $m = 12$ satisfies the equation: $\frac{n^2}{12} = \frac{100}{12} = 10$, $m = 12$, so $m + n = 22$.
Correct Answer: 22