If $a, b$ are positive real numbers such that the lines $a x+9 y=5$ and $4 x+b y=3$ are parallel, then the least possible value of $a +b$ is

Let $f: R \rightarrow R$ be such that for all $\mathrm{x} \in \mathrm{R}\left(2^{1+\mathrm{x}}+2^{1-\mathrm{x}}\right), f(\mathrm{x})$ and $\left(3 ^\mathrm{x}+3^{-\mathrm{x}}\right)$ are in $A.P.$, then the minimum value of $f(x)$ is

If $a_1, a_2...,a_n$ an are positive real numbers whose product is a fixed number $c$ , then the minimum value of $a_1 + a_2 +.... + a_{n - 1} + 2a_n$ is

Let $\mathrm{A}_1, \mathrm{G}_1, \mathrm{H}_1$ denote the arithmetic, geometric and harmonic means, respectively, of two distinct positive numbers. For $\mathrm{n} \geq 2$, let $A_{n-1}$ and $H_{n-1}$ has arithmetic, geometric and harmonic means as $A_n, G_n, H_n$ respectively.

$1.$  Which one of the following statements is correct?

$(A)$ $\mathrm{G}_1>\mathrm{G}_2>\mathrm{G}_3>\ldots$

$(B)$ $\mathrm{G}_1<\mathrm{G}_2<\mathrm{G}_3<\ldots$

$(C)$ $\mathrm{G}_1=\mathrm{G}_2=\mathrm{G}_3=\ldots$

$(D)$ $\mathrm{G}_1<\mathrm{G}_3<\mathrm{G}_5<\ldots$ and $\mathrm{G}_2>\mathrm{G}_4>\mathrm{G}_6>\ldots$

$2.$ Which of the following statements is correct?

$(A)$ $A_1>A_2>A_5>\ldots$

$(B)$ $A_1$

$(C)$ $\mathrm{A}_1>\mathrm{A}_3>\mathrm{A}_5>\ldots$ and $\mathrm{A}_2<\mathrm{A}_4<\mathrm{A}_6<\ldots$

$(D)$ $A_1A_4 > A_6 > \ldots$

$3.$ Which of the following statements is correct?

$(A)$ $\mathrm{H}_1>\mathrm{H}_2>\mathrm{H}_3>\ldots$

$(B)$ $\mathrm{H}_1<\mathrm{H}_2<\mathrm{H}_3<\ldots$

$(C)$ $\mathrm{H}_1>\mathrm{H}_3>\mathrm{H}_5>\ldots$ and $\mathrm{H}_2<\mathrm{H}_4<\mathrm{H}_6<\ldots$

$(D)$ $\mathrm{H}_1<\mathrm{H}_3<\mathrm{H}_5<\ldots$ and $\mathrm{H}_2>\mathrm{H}_4>\mathrm{H}_6>\ldots$

Give the answer question $1,2$ and $3.$

If $a$ be the arithmetic mean of $b$ and $c$ and ${G_1},\;{G_2}$ be the two geometric means between them, then $G_1^3 + G_2^3 = $

If $a,\;b,\;c$ are in $G.P.$ and $\log a - \log 2b,\;\log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$, then $a,\;b,\;c$ are the length of the sides of a triangle which is