If $a,\;b,\;c$ are in $A.P.$ and $a,\;b,\;d$ in $G.P.$, then $a,\;a - b,\;d - c$ will be in

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 each term of a geometric progression $a_1, a_2, a_3, \ldots$ with $a_1=\frac{1}{8}$ and $a_2 \neq a_1$, is the arithmetic mean of the next two terms and $S_n=a_1+a_2+\ldots+a_n$, then $\mathrm{S}_{20}-\mathrm{S}_{18}$ is equal to

Given a sequence of $4$ numbers, first three of which are in $G.P.$ and the last three are in $A.P$. with common difference six. If first and last terms in this sequence are equal, then the last term is

If $A.M.$ of two terms is $9$ and $H.M.$ is $36$, then $G.M.$ will be

If $x\in (0,\frac{\pi}{4})$ then the expression $ \frac{cos x}{sin^2 x(cos x-sin x)}$ can not take the value