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1. $\frac{1}{1 + a^{(n - m)}} + \frac{1}{1 + a^{(m - n)}} = ?$

A. 0

B. $\frac{1}{2}$

C. 1

D. a^{m + n}

Solution:

\begin{aligned} \frac{1}{1 + a^{(n - m)}} + \frac{1}{1 + a^{(m - n)}} \\ = \frac{1}{(1 + \frac{a^{n}}{a^{m}})} + \frac{1}{(1 + \frac{a^{m}}{a^{n}})} \\ = \frac{a^{m}}{(a^{m} + a^{n})} + \frac{a^{n}}{(a^{m} + a^{n})} \\ = \frac{(a^{m} + a^{n})}{(a^{m} + a^{n})} \\ = 1 \end{aligned}

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