Large Exponents

We apply jensen's inequality .

Since the logarithmic function $f(x) = \log x$ is concave, we know that:

$\sum_{i=1}^n \log a_i \leq n \times \log \left ( \frac { \sum _{i=1}^n a_i } { n } \right) .$

Taking the exponent, this implies that $\prod_{i=1}^n a_i \leq \left( \frac{ \sum_{i=1}^n a_i } { n } \right)^n ,$

and equality holds only if all of the terms are equal.

Applying this to $a_1 = a_2 = \ldots = a_{2014} = 2017, a_{2015} = a_{2016} = 1$ , we conclude that:

$2017^{2014} = 2017^{2014} \times 1 \times 1 < \left(\frac{ 2017 \times 2014 + 1 + 1 } { 2016} \right) ^ { 2016 } = 2015^{2016}$

Clearly, these quantities are not equal since all powers of 2015 will terminate with a 5 while powers of 2017will terminate with 1, 3, 7, or 9. If we take two division cases:

CASE I: $\frac{2015^{2016}}{2017^{2014}} = (\frac{2015}{2017})^{2014} \cdot 2015^2 \approx 550,583$

CASE II: $\frac{2017^{2014}}{2015^{2016}} = (\frac{2017}{2015})^{2014} \cdot 2015^{-2} \approx 0$

Hence, choice B is correct.