If $A_1, A_2$ are two arithmetic means,$G_1, G_2$ are two geometric means,and $H_1, H_2$ are two harmonic means between two numbers,then $\frac{A_1 + A_2}{H_1 + H_2} \cdot \frac{H_1 H_2}{G_1 G_2} = \dots$

Let $a_{1}=1$ and for $n \ge 1$, $a_{n+1} = \frac{1}{2}a_{n} + \frac{n^{2}-2n-1}{n^{2}(n+1)^{2}}$. Then $|\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})|$ is equal to ........... .

Let $S_{n}(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \ldots$ up to $n$-terms,where $a > 1$. If $S_{24}(x) = 1093$ and $S_{12}(2x) = 265$,then the value of $a$ is equal to ..... .

Let $3, 7, 11, 15, \ldots, 403$ and $2, 5, 8, 11, \ldots, 404$ be two arithmetic progressions. Then the sum of the common terms in them is equal to:

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

Two sequences $\{t_n\}$ and $\{s_n\}$ are defined by $t_n = \log \left( \frac{5^{n+1}}{3^{n-1}} \right)$ and $s_n = \left[ \log \left( \frac{5}{3} \right) \right]^n$. Then: