If the numbers $a, b, c, d, e$ form an $A.P.$,then the value of $a - 4b + 6c - 4d + e$ is

The sum of the series $1 \cdot 3^2 + 2 \cdot 5^2 + 3 \cdot 7^2 + \dots$ up to $20$ terms is

Let $S$ be the sum of the first $9$ terms of the series: $(x+ka) + (x^2+(k+2)a) + (x^3+(k+4)a) + (x^4+(k+6)a) + \dots$ where $a \neq 0$ and $x \neq 1$. If $S = \frac{x^{10}-x+45a(x-1)}{x-1}$,then $k$ is equal to:

If $x_1, x_2, \dots, x_n$ and $\frac{1}{h_1}, \frac{1}{h_2}, \dots, \frac{1}{h_n}$ are two $A.P.s$ such that $x_3 = h_2 = 8$ and $x_8 = h_7 = 20$,then $x_5 \cdot h_{10}$ equals

Find the sum of an $AP$ of $40$ terms whose first and last terms are $80$ and $275$.

Find the sum of $1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots$ to infinite terms.