The value of $0.2overline{34}$ is

Let ${a_n}$ be the ${n^{th}}$ term of the $G$.$P$. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $,such that $\alpha \ne \beta $,then the common ratio is

If three distinct numbers $a, b, c$ are in $G.P.$ and the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root,then which one of the following statements is correct?

Find the sum of all integers between $50$ and $500$ which are divisible by $7.$

Maximum value of the sum of the Arithmetic Progression $50, 48, 46, 44, \dots$ is

If $a, b, c, d$ are in $H.P.$,then