If $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \dots \infty$,then the value of $\alpha$ is:

If $S = \frac{7}{5} + \frac{9}{5^{2}} + \frac{13}{5^{3}} + \frac{19}{5^{4}} + \ldots$, then $160 \,S$ is equal to....... .

Evaluate the sum: $\sum\limits_{k = 1}^n {k\left( {1 + \frac{1}{n}} \right)^{k - 1}}$

Let $p(x)=a_0+a_1 x+\ldots+a_n x^n$. If $p(-2)=-15, p(-1)=1, p(0)=7, p(1)=9, p(2)=13$ and $p(3)=25$,then the smallest possible value of $n$ is

If $\tan \theta = \sqrt{\frac{3}{2}}$,the sum of the infinite series $1 + 2(1 - \cos \theta) + 3(1 - \cos \theta)^2 + 4(1 - \cos \theta)^3 + \dots \infty$ is

Let $f(x) = \sin x + 2\sin^2 x + 3\sin^3 x + 4\sin^4 x + \dots \infty$. Then the number of solutions of the equation $f(x) = 2$ in $x \in [-\pi, \pi] - \{\pm \frac{\pi}{2}\}$ is