If three unequal numbers $p, q, r$ are in $H.P.$ and their squares are in $A.P.$,then the ratio $p:q:r$ is

For all $n \in N$,which of the following is true: $\frac{3^n-1}{2} \geq$ ?

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

Let four distinct positive numbers $a_1, a_2, a_3, a_4$ be in a geometric progression. Let $b_1 = a_1$,$b_2 = b_1 + a_2$,$b_3 = b_2 + a_3$,and $b_4 = b_3 + a_4$.
Statement-$I$: The numbers $b_1, b_2, b_3, b_4$ are neither in an arithmetic progression nor in a geometric progression.
Statement-$II$: The numbers $b_1, b_2, b_3, b_4$ are in a harmonic progression.

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

If $a, b, c$ are in $G.P.$ and $\log a - \log 2b, \log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$,then $a, b, c$ are the lengths of the sides of a triangle which is