Sequences and Series

$S_n=1 \cdot 3+2 \cdot 2^2+3 \cdot 3^3+4 \cdot 2^4+\ldots \ldots \ldots$ upto $n$ terms. If $S_{20}=a \cdot 3^{21}+b \cdot 2^{22}+\frac{391}{288}$, then value of $32 a-9 b$ is

Sum to 10 terms of the series $1+2(1 \cdot 1)+3(1 \cdot 1)^2+4(1 \cdot 1)^3+$ $\_\_\_\_$ is

The sum of the series $1 \cdot 2^2+2 \cdot 4^2+3 \cdot 6^2+\ldots$ upto 10 terms is

The sum of $1+\frac{1}{4}+\frac{1 \cdot 3}{4 \cdot 8}+\frac{1 \cdot 3 \cdot 5}{4 \cdot 8 \cdot 12}+\ldots \infty$ is

Sum of first ' $n$ ' terms of a series $a_1+a_2+\ldots+a_n$ is given by $S_n=\frac{n\left(n^2-1\right)(n+2)}{4}$, then the value of $\lim _\limits{n \rightarrow \infty} \sum_\limits{r=2}^n \frac{1}{a_r}$ is

The value of \(\frac{1}{2}\left(\frac{1}{5}\right)^2+\frac{2}{3}\left(\frac{1}{5}\right)^3+\frac{3}{4}\left(\frac{1}{5}\right)^4+\ldots . . \infty\) is

Let \(a_n\) be a sequence of numbers which is defined by relation \(a_1=2, \frac{a_n}{a_{n+1}}=3^{-n}\), then \(\log _2\left(a_{50}\right)\) is equal to (take \(\log _2 3=1.6\) )

If the real numbers \(x, y, z, t\) be in GP then the value of \((x^2+y^2+z^2)(y^2+z^2+t^2)\) is euqal to

The value of \(\frac{4}{1 !}+\frac{11}{2 !}+\frac{22}{3 !}+\frac{37}{4 !}+\frac{56}{5 !}+\ldots \infty\) is