Sum the series $51 + 50 + 49 + \ldots + 21$.

If the $10^{\text{th}}$ term of an $A$.$P$. is $\frac{1}{20}$ and its $20^{\text{th}}$ term is $\frac{1}{10}$,then the sum of its first $200$ terms is

Let $\alpha$ and $\beta$ be the roots of $x^{2}-3x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^{2}-6x+q=0$. If $\alpha, \beta, \gamma, \delta$ form a geometric progression,then the ratio $(2q+p):(2q-p)$ is:

If $\frac{S_n}{S_m} = \frac{n^4}{m^4}$ (where $S_k$ is the sum of the first $k$ terms of an $A$.$P$. $a_1, a_2, \dots, \infty$),then the value of $\frac{a_{m+1}}{a_{n+1}}$ in terms of $m$ and $n$ will be

If $a_1, a_2, a_3, \dots$ are in $A.P.$ such that $a_1 + a_7 + a_{16} = 40$, then the sum of the first $15$ terms of this $A.P.$ is

The sum of $100$ terms of the series $0.9 + 0.09 + 0.009 + \dots$ will be