If $\log 2,\;\log ({2^n} – 1)$ and $\log ({2^n} + 3)$ are in $A.P.$, then $n =$

The $A.M.$ of a $50$ set of numbers is $38$. If two numbers of the set, namely $55$ and $45$ are discarded, the $A.M.$ of the remaining set of numbers is

If $x_1 , x_2 ,  ….. , x_n$ and $\frac{1}{{{h_1}}},\frac{1}{{{h^2}}},……\frac{1}{{{h_n}}}$ are two $A.P' s$ such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$, then $x_5. h_{10}$ equals

Let $a_1, a_2 , a_3,…..$ be an $A.P$, such that $\frac{{{a_1} + {a_2} + …. + {a_p}}}{{{a_1} + {a_2} + {a_3} + ….. + {a_q}}} = \frac{{{p^3}}}{{{q^3}}};p \ne q$. Then $\frac{{{a_6}}}{{{a_{21}}}}$ is equal to

If the roots of the equation $x^3 – 9x^2 + \alpha x – 15 = 0 $ are in $A.P.$, then $\alpha$  is