Let $a, b, c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^5 b^3 c^2 d$ is $3750 \beta$,then the value of $\beta$ is

The first term of an arithmetic progression is $10$ and the last term is $50$. If the sum of all its terms is $300$,then the number of terms $n = ...$

If the $7^{th}$ term of an $A.P.$ is $40$,then the sum of the first $13$ terms is:

Let $a_1, a_2, a_3, \ldots$ be terms of an $A.P.$ If $\frac{a_1 + a_2 + \ldots + a_p}{a_1 + a_2 + \ldots + a_q} = \frac{p^2}{q^2}$ for $p \ne q$,then $\frac{a_6}{a_{21}}$ equals:

If $\frac{3 + 5 + 7 + \dots \text{ to } n \text{ terms}}{5 + 8 + 11 + \dots \text{ to } 10 \text{ terms}} = 7$,then the value of $n$ is

If the $9^{th}$ and $19^{th}$ terms of an arithmetic progression are $35$ and $75$ respectively,what is its $20^{th}$ term?