Sequence of series practice · Hard

For $x \in R, x \neq-1$, if $(1+x)^{2016}+x(1+x)^{2015}+x^{2}$ $(1+x)^{2014}+\ldots .+\mathrm{x}^{2016}=\sum_{\mathrm{i}=0}^{2016} \mathrm{a}_{\mathrm{i}} \mathrm{x}^{\mathrm{i}}$, then $\mathrm{a}_{17}$ is equal to $:$

Suppose $a,\;b,\;c$ are in AP and ${a^2},\;{b^2},\;{c^2}$ are in GP. If $a < b < c$ and $a + b + c = \frac{3}{2}$, then the value of $a$ is

The sum of first $n$ terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\ldots$ is

The sum of all the products of the first $n$ natural numbers taken two at a time is

The sum of first 20 terms of $3 \cdot 5+7 \cdot 5^2+11 \cdot 5^3+\ldots$ is

If $a,\;b,\;c,\;d\;\mathrm{and}\;p$ are different real numbers such that $\left( {{a^2} + {b^2} + {c^2}} \right)\;{p^2} - 2\left( {ab + bc + cd} \right)\;\;p + \left( {{b^2} + {c^2} + {d^2}} \right) \le 0,\mathrm{then}\;a,\;b,\;c,\;d$ are in

Let ${S_1},\;{S_2}, \ldots$ 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 10cm, then for which of the following values of $n$ is the area of ${S_n}$ less than 1 sq cm?