$(1^{2}+2^{2}+3^{2}+\cdots+10^{2})$ is equal to:

The sum $1(1!) + 2(2!) + 3(3!) + ....+n (n!)$ equals

The natural numbers are written as follows. The sum of the numbers in the $n^{th}$ row is:
$1$
$2$ $3$
$4$ $5$ $6$
$7$ $8$ $9$ $10$
$. . .$

Let $a, b$ and $c$ be the $7^{th}, 11^{th}$ and $13^{th}$ terms respectively of a non-constant $A.P.$ If these are also the three consecutive terms of a $G.P.$,then $\frac{a}{c}$ is equal to

If $a_1, a_2, ..., a_n$ are positive real numbers whose product is a fixed number $c$,then the minimum value of $a_1 + a_2 + ... + a_{n-1} + 2a_n$ is

If $\omega$ is a non-real root of the equation $x^3 - 1 = 0$,then the value of $\sum_{r=1}^5 (1 + \omega^r + \omega^{2r})$ is