CAT (QA) PYQs

In an arithmetic progression, if the sum of fourth, seventh and tenth terms is 99, and the sum of the first fourteen terms is 497, then the sum of first five terms is

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

If n is a positive integer such that $(\sqrt[7]{10})(\sqrt[7]{10})^{2} \ldots(\sqrt[7]{10})^{n}>999$, then the smallest value of n is