Find the sum of the first $n$ terms of the $AP$ $\frac{n-1}{n}, \frac{n+1}{n}, \frac{n+2}{n}, \frac{n+3}{n}, \cdots$

If $3 + \frac{1}{4} (3 + d) + \frac{1}{4^2} (3 + 2d) + \dots \infty = 8$,then the value of $d$ is:

The harmonic mean of two numbers is $4$ and the arithmetic and geometric means satisfy the relation $2A + G^2 = 27$. The numbers are:

$\frac{{\frac{1}{2} \cdot \frac{2}{2}}}{{{1^3}}} + \frac{{\frac{2}{2} \cdot \frac{3}{2}}}{{{1^3} + {2^3}}} + \frac{{\frac{3}{2} \cdot \frac{4}{2}}}{{{1^3} + {2^3} + {3^3}}} + \dots + n \text{ terms} =$

The geometric series $a + ar + ar^2 + ar^3 + \dots \infty$ has a sum of $7$,and the sum of the terms involving odd powers of $r$ is $3$. Then,the value of $(a^2 - r^2)$ is:

Let $\frac{1}{x_1}, \frac{1}{x_2}, \frac{1}{x_3}, \dots, \frac{1}{x_n}$ ($x_i \neq 0$ for $i = 1, 2, \dots, n$) be in $A.P.$ such that $x_1 = 4$ and $x_{21} = 20$. If $n$ is the least positive integer for which $x_n > 50$,then $\sum_{i=1}^n \frac{1}{x_i}$ is equal to: