If $A$ is the $A.M.$ of the roots of the equation ${x^2} - 2ax + b = 0$ and $G$ is the $G.M.$ of the roots of the equation ${x^2} - 2bx + {a^2} = 0,$ then

If the ratio of two numbers be $9:1$, then the ratio of geometric and harmonic means between them will be

Let $a_{1}, a_{2}, \ldots, a_{10}$ be an $AP$ with common difference $-3$ and $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{b}_{10}$ be a $GP$ with common ratio $2.$ Let $c_{k}=a_{k}+b_{k}, k=1,2, \ldots, 10 .$ If $c_{2}=12$ and $\mathrm{c}_{3}=13$, then $\sum_{\mathrm{k}=1}^{10} \mathrm{c}_{\mathrm{k}}$ is equal to ..... .

Let the sum of an infinite $G.P.$, whose first term is $a$ and the common ratio is $r$, be $5$. Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first $21$ terms of an $AP$, whose first term is $10\,ar , n ^{\text {th }}$ term is $a_{n}$ and the common difference is $10{a r^{2}}$, is equal to.

If the arithmetic and geometric means of $a$ and $b$ be $A$ and $G$ respectively, then the value of $A - G$ will be

Let $\frac{1}{16}, a$ and $b$ be in $G.P.$ and $\frac{1}{ a }, \frac{1}{ b }, 6$ be in $A.P.,$ where $a , b >0$. Then $72( a + b )$ is equal to ...... .