If all the terms of an $A.P.$ are squared, then new series will be in

If $a,\;b,\,c$ be in $G.P.$ and $a + x,\;b + x,\;c + x$ in $H.P.$, then the value of $x$ is ($a,\;b,\;c$ are distinct numbers)

If three unequal non-zero real numbers $a,\;b,\;c$ are in $G.P.$ and $b - c,\;c - a,\;a - b$ are in $H.P.$, then the value of $a + b + c$ is independent of

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

If $a, b, c$ are in $A.P.;$ $b, c, d$ are in $G.P.$ and $\frac{1}{c}, \frac{1}{d}, \frac{1}{e}$ are in $A.P.$ prove that $a, c, e$ are in $G.P.$

If the product of three terms of $G.P.$ is $512$. If $8$ added to first and $6$ added to second term, so that number may be in $A.P.$, then the numbers are