The solution of the equation $1 + a + a^2 + a^3 + \dots + a^x = (1 + a)(1 + a^2)(1 + a^4)$ is given by $x$ is equal to

$a, g, h$ are the arithmetic mean,geometric mean,and harmonic mean between two positive numbers $x$ and $y$ respectively. Identify the correct statement among the following:

Find the sum of infinite terms of the series $1+\frac{2}{3}+\frac{3}{3^{2}}+\frac{4}{3^{3}}+\frac{5}{3^{4}}+\cdots$

In a geometric progression,if the ratio of the sum of first $5$ terms to the sum of their reciprocals is $49$,and the sum of the first and the third term is $35$. Then the first term of this geometric progression is

If $\log 2, \log (2^n - 1)$ and $\log (2^n + 3)$ are in $A.P.$,then $n =$

The $n^{th}$ term of the series $\frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \dots$ will be