Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. Then the greatest value of $n$ for which $a_n$ is the greatest is

If the value of $\left(1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\ldots \text{ to } \infty\right)^{\log_{(0.25)}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots \text{ to } \infty\right)}$ is $l$,then $l^{2}$ is equal to $......$

Let $I(n) = n^n$ and $J(n) = 1 \times 3 \times 5 \times \ldots \times (2n - 1)$ for all $n > 1, n \in N$. Then:

Three unequal positive numbers $a, b, c$ are such that $a, b, c$ are in $G.P.$ while $\log \left(\frac{5 c}{2 a}\right), \log \left(\frac{7 b}{5 c}\right), \log \left(\frac{2 a}{7 b}\right)$ are in $A.P.$ Then $a, b, c$ are the lengths of the sides of

If $e^{(\cos^{2} x + \cos^{4} x + \cos^{6} x + \dots \infty) \log_{e} 2}$ satisfies the equation $t^{2} - 9t + 8 = 0$,then the value of $\frac{2 \sin x}{\sin x + \sqrt{3} \cos x}$ for $0 < x < \frac{\pi}{2}$ is

Write the first five terms of the sequence whose $n^{th}$ term is $a_{n} = (-1)^{n-1} 5^{n+1}$.