The value of $x$ satisfying ${\log _a}x + {\log _{\sqrt a }}x + {\log _{3\sqrt a }}x + .........{\log _{a\sqrt a }}x = \frac{{a + 1}}{2}$ will be

The number of $5 -$tuples $(a, b, c, d, e)$ of positive integers such that

$I.$ $a, b, c, d, e$ are the measures of angles of a convex pentagon in degrees

$II$. $a \leq b \leq c \leq d \leq e$

$III.$ $a, b, c, d, e$ are in arithmetic progression is

If ${a_1},\,{a_2},....,{a_{n + 1}}$ are in $A.P.$, then $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + ..... + \frac{1}{{{a_n}{a_{n + 1}}}}$ is

Find the $7^{\text {th }}$ term in the following sequence whose $n^{\text {th }}$ term is $a_{n}=\frac{n^{2}}{2^{n}}$

If the sum of three numbers in $A.P.,$ is $24$ and their product is $440,$ find the numbers.

How many terms of the $A.P.$ $-6,-\frac{11}{2},-5, \ldots \ldots$ are needed to give the sum $-25 ?$