CAT (QA) PYQs

If $a_{1}, a_{2} \ldots .$. are in A.P., then, $\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\ldots .+\frac{1}{\sqrt{a_{n}}+\sqrt{a_{n+1}}}$ is equal to

The natural numbers are divided into groups as (1), $(2,3,4),(5,6,7,8,9), \ldots .$. and so on. Then, the sum of the numbers in the 15th group is equal to

The sum of the infinite series is $\frac{1}{5}\left(\frac{1}{5}-\frac{1}{7}\right)+\left(\frac{1}{5}\right)^{2}\left(\left(\frac{1}{5}\right)^{2}-\left(\frac{1}{7}\right)^{2}\right)+\left(\frac{1}{5}\right)^{3}\left(\left(\frac{1}{5}\right)^{3}-\left(\frac{1}{7}\right)^{3}\right)+\ldots .$. equal to