In a $G.P.$ the sum of three numbers is $14$, if $1 $ is added to first two numbers and subtracted from third number, the series becomes $A.P.$, then the greatest number is

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 ..... .

The reciprocal of the mean of the reciprocals of $n$ observations is their

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 ...... .

If $\frac{{a + bx}}{{a - bx}} = \frac{{b + cx}}{{b - cx}} = \frac{{c + dx}}{{c - dx}}(x \ne 0)$, then $a,\;b,\;c,\;d$ are in

If $a,\;b,\;c$ are in $A.P.$ and $a,\;c - b,\;b - a$ are in $G.P. $ $(a \ne b \ne c)$, then $a:b:c$ is