$\mathrm{S}_1=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}, \mathrm{S}_2=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2$ and $\mathrm{S}_3=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^3$, then the value of $\lim _\limits{n \rightarrow \infty} \frac{s_1\left(1+\frac{s_3}{4}\right)}{s_2^2}$ is

If $$x, y, z$$ are in A.P. and $$\tan ^{-1} x, \tan ^{-1} y$$ and $$\tan ^{-1} z$$ are also in A.P., then

If for an arithmetic progression, 9 times nineth term is equal to 13 times thirteenth term, then value of twenty second term is

For a sequence $\left(t_n\right)$, if $S_n=5\left(2^n-1\right)$ then $t_n=$ .........

If $A, B, C$ are $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a GP respectively then $A^{q-r} \cdot B^{r-p} \cdot C^{p-q}=\ldots \ldots$

If $\sum_{r=1}^n(2 r+1)=440$, then $n=$ ...............

If the sum of an infinite GP be 9 and sum of first two terms be 5 then their common ratio is ..........

For a GP, if $S_n=\frac{4^n-3^n}{3^n}$, then $t_2=$ ...........

For a GP, if $(m+n)^{\text {th }}$ term is $p$ and $(m-n)^{\mathrm{th}}$ term is $q$, then $m^{\text {th }}$ term is ......