If $a, b, c$ are in both Arithmetic Progression $(A.P.)$ and Geometric Progression $(G.P.)$,then......

An $A.P.$,a $G.P.$,and a $H.P.$ have the same first and last terms and the same odd number of terms. The middle terms of the three series are in

If $a$ and $b$ are the roots of $x^{2}-3x+p=0$ and $c$ and $d$ are roots of $x^{2}-12x+q=0$,where $a, b, c, d$ form a $G.P.$,prove that $(q+p):(q-p)=17:15$.

Given a sequence of $4$ numbers,the first three of which are in $G.P.$ and the last three are in $A.P.$ with a common difference of $6$. If the first and last terms in this sequence are equal,then the last term is:

If the arithmetic mean and geometric mean of two positive numbers are $A$ and $G$ respectively,then the numbers are ..........

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