Insert $6$ arithmetic means between $20$ and $76$.

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a given $A.P.$ whose common difference is an integer and $S_{n} = a_{1} + a_{2} + \ldots + a_{n}$. If $a_{1} = 1$,$a_{n} = 300$ and $15 \leq n \leq 50$,then the ordered pair $(S_{n-4}, a_{n-4})$ is equal to

If $x \in (0, \frac{\pi}{4})$,then the expression $\frac{\cos x}{\sin^2 x(\cos x - \sin x)}$ cannot take the value

In a $G.P.$,the sum of three numbers is $14$. If $1$ is added to the first two numbers and subtracted from the third number,the series becomes an $A.P.$. Find the greatest number.

If $\sum_{i = 1}^n \sum_{j = 1}^i \sum_{k = 1}^j 1 = 560$,then the value of $n$ is:

Three numbers are selected from the set ${3^1, 3^2, 3^3, \dots, 3^{20}}$. Find the number of ways these selected numbers can form an increasing Geometric Progression $(G.P.)$.