Let $\left\langle a_n\right\rangle$ be a sequence such that $a_0=0, a_1=\frac{1}{2}$ and $2 a_{n+2}=5 a_{n+1}-3 a_n, n=0,1,2,3, \ldots \ldots$. Then $\sum_{k=1}^{100} a_k$ is equal to :

How many terms of the $G.P.$ $3, \frac{3}{2}, \frac{3}{4}, \ldots$ are needed to give the sum $\frac{3069}{512} ?$

If $1 + \cos \alpha + {\cos ^2}\alpha + .......\,\infty = 2 - \sqrt {2,} $ then $\alpha ,$ $(0 < \alpha < \pi )$ is

The sum of first two terms of a $G.P.$ is $1$ and every term of this series is twice of its previous term, then the first term will be

The value of ${a^{{{\log }_b}x}}$, where $a = 0.2,\;b = \sqrt 5 ,\;x = \frac{1}{4} + \frac{1}{8} + \frac{1}{{16}} + .........$to $\infty $ is

Find the sum of the following series up to n terms:

$5+55+555+\ldots$