If $\frac{{b + a}}{{b - a}} = \frac{{b + c}}{{b - c}}$, then $a,\;b,\;c$ are in

If $a,\,b,\;c$ are in $A.P.$ and ${a^2},\;{b^2},\;{c^2}$ are in $H.P.$, then

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 $3.$

The harmonic mean of two numbers is $4$ and the arithmetic and geometric means satisfy the relation $2A + {G^2} = 27$, the numbers are

The sum of three consecutive terms in a geometric progression is $14$. If $1$ is added to the first and the second terms and $1$ is subtracted from the third, the resulting new terms are in arithmetic progression. Then the lowest of the original term is

If $A.M.$ and $G.M.$ of roots of a quadratic equation are $8$ and $5,$ respectively, then obtain the quadratic equation.