If the $p^{th}$,$q^{th}$,and $r^{th}$ terms of an $A.P.$ are $a, b,$ and $c$ respectively,find the value of $a(q-r) + b(r-p) + c(p-q)$.

The sum of the first $n$ terms of an $A.P.$ is given by $S_n = 2n + 3n^2$. Another $A.P.$ is formed with the same first term and double the common difference. The sum of the first $n$ terms of this new $A.P.$ is:

Consider a sequence whose sum of first $n$ terms is given by $S_n = 4n^2 + 6n$,where $n \in N$. Find the $15^{th}$ term $(T_{15})$ of this sequence.

The sum of the first $n$ terms of a sequence is given by $S_n = 3n^2 + 4n + 15$. If $T_r$ is the $r^{th}$ term of the sequence,then $T_3 - T_1$ is equal to:

Sum of the series $1 \cdot 2015 + 2 \cdot 2014 + 3 \cdot 2013 + \dots + 2015 \cdot 1$ is equal to :-

The sum of the first $20$ terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots$ is?