\displaystyle \frac{2^k5^n + 3^k2^n}{7^k5^n + 10^k3^n} = \frac{2^k + 3^k\left(\frac{2}{5}\right)^n}{7^k + 10^k\left(\frac{3}{5}\right)^n} \leq \frac{2^k + 3^k}{7^k + 10^k} \leq \frac{2\cdot 3^k}{2\cdot 7^k} = \left(\frac{3}{7}\right)^k