For what value of m, $${{{a^{m + 1}} + {b^{m + 1}}} \over {{a^m} + {b^m}}}$$ is the arithmetic mean of a and b?

The coefficient of xⁿ, where n is any positive integer, in the expansion of (1 + 2x + 3x² + ....... $$\infty$$)^1/2 is

The sum of the infinite series $$1 + {1 \over {2!}} + {{1\,.\,3} \over {4!}} + {{1\,.\,3\,.\,5} \over {6!}} + \,....$$ is

If sum of an infinite geometric series is $${4 \over {3}}$$ and its 1^st term is $${3 \over {4}}$$, then its common ratio is