The sum of the rational terms in the expansion of $${\left( {\sqrt 2 + {3^{1/5}}} \right)^{10}}$$ is ...............

Let $$n$$ be positive integer. If the coefficients of 2nd, 3rd, and 4th terms in the expansion of $${\left( {1 + x} \right)^n}$$ are in A.P., then the value of $$n$$ is ................

If $${\left( {1 + ax} \right)^n} = 1 + 8x + 24{x^2} + .....$$ then $$a=..........$$ and $$n =............$$

The larger of $${99^{50}} + {100^{50}}$$ and $${101^{50}}$$ is ..............