Harmonic mean of the roots of the equation $(5 + \sqrt{2})x^2 - (4 + \sqrt{5})x + 8 + 2\sqrt{5} = 0$ is:

Let $a_1, a_2, \dots, a_{49}$ be in $A.P.$ such that $\sum_{k=0}^{12} a_{4k+1} = 416$ and $a_9 + a_{43} = 66$. If $a_1^2 + a_2^2 + \dots + a_{17}^2 = 140m$,then $m = \dots$

The $G.M.$ of the roots of the equation $x^2 - 18x + 9 = 0$ is

If the sum $\frac{3}{1^2} + \frac{5}{1^2 + 2^2} + \frac{7}{1^2 + 2^2 + 3^2} + \dots$ up to $20$ terms is equal to $\frac{k}{21}$,then $k$ is equal to

The value of the expression $3(1!) - 4(2!) + 5(3!) - 6(4!) + \dots - 2008(2006)! + (2007)!$ is:

Let $a_1, a_2, a_3, \dots, a_{100}$ be positive real numbers and $S_k$ be the sum of products of $a_1, a_2, \dots, a_{100}$ taken $k$ at a time. If $S_{98} S_2 \ge \lambda (a_1 a_2 \dots a_{100})$,then $\lambda$ is