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

In a $G.P.$,the first term is $7$,the last term is $448$,and the sum is $889$. Find the common ratio.

If $T_n = (n^2 + 1)n!$ and $S_n = T_1 + T_2 + T_3 + ...... + T_n$. Let $\frac{T_{10}}{S_{10}} = \frac{a}{b}$,where $a$ and $b$ are relatively prime natural numbers,then the value of $(b - a)$ is:

If the $7^{th}$ term of a harmonic progression is $8$ and the $8^{th}$ term is $7$,then its $15^{th}$ term is

The interior angles of an $n$-sided convex polygon are in $G.P.$ The smallest angle is $120^\circ$ and the common ratio is $2$. Find the number of possible values of $n$.

There are two such pairs of non-zero real values of $a$ and $b$,i.e.,$(a_1, b_1)$ and $(a_2, b_2)$,for which $2a+b, a-b, a+3b$ are three consecutive terms of a $G.P.$. Then the value of $2(a_1b_2 + a_2b_1) + 9a_1a_2$ is-