If the $(m + 1)^{th}$,$(n + 1)^{th}$,and $(r + 1)^{th}$ terms of an arithmetic progression are in geometric progression,and $m, n, r$ are in harmonic progression,then what is the ratio of the common difference to the first term of the arithmetic progression?

Three numbers are in an increasing geometric progression with common ratio $r$. If the middle number is doubled,then the new numbers are in an arithmetic progression with common difference $d$. If the fourth term of the $G.P.$ is $3r^{2}$,then $r^{2}-d$ is equal to:

Let $n \geq 4$ be a positive integer and let $l_1, l_2, \ldots, l_n$ be the lengths of the sides of an arbitrary $n$-sided non-degenerate polygon $P$. Suppose $\frac{l_1}{l_2} + \frac{l_2}{l_3} + \ldots + \frac{l_{n-1}}{l_n} + \frac{l_n}{l_1} = n$. Consider the following statements:
$I$. The lengths of the sides of $P$ are equal.
$II$. The angles of $P$ are equal.
$III$. $P$ is a regular polygon if it is cyclic.

Let $a = \min \{x^{2} + 2x + 3 : x \in R\}$ and $b = \lim_{\theta \rightarrow 0} \frac{1 - \cos \theta}{\theta^{2}}$. Then $\sum_{r=0}^{n} a^{r} b^{n-r}$ is

Write the first five terms of the following sequence and obtain the corresponding series:
$a_{1} = a_{2} = 2, a_{n} = a_{n-1} - 1, n > 2$

Let $\{a_k\}$ and $\{b_k\}, k \in N$,be two $G$.$P$.s with common ratios $r_1$ and $r_2$ respectively such that $a_1=b_1=4$ and $r_1 < r_2$. Let $c_k=a_k+b_k, k \in N$. If $c_2=5$ and $c_3=13/4$,then $\sum_{k=1}^{\infty} c_k - (12a_6 + 8b_4)$ is equal to