If $\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3 = \frac{k}{21}$,then $k$ equals

If the $7^{th}$ term of a harmonic progression is $8$ and the $8^{th}$ term is $7$,then its $15^{th}$ term is

If the sum of an infinite $G.P.$ and the sum of the squares of its terms is $3$,then the common ratio of the first series is

Let $S_1, S_2, \dots$ be squares such that for each $n \ge 1$,the length of a side of $S_n$ equals the length of a diagonal of $S_{n+1}$. If the length of a side of $S_1$ is $10 \text{ cm}$,then for which of the following values of $n$ is the area of $S_n$ less than $1 \text{ cm}^2$?

If $\sum_{n = 1}^5 \frac{1}{n(n + 1)(n + 2)(n + 3)} = \frac{k}{3}$,then $k$ is equal to

If $x = \frac{4}{3} - \frac{4x}{9} + \frac{4x^2}{27} - \dots \infty$,then $x$ is equal to