Find the value of $n$ so that $\frac{a^{n+1}+b^{n+1}}{a^{n}+b^{n}}$ may be the geometric mean between $a$ and $b$.

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).

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

The arithmetic mean and the geometric mean of two distinct $2$-digit numbers $x$ and $y$ are two integers,one of which can be obtained by reversing the digits of the other (in base $10$ representation). Then,$x+y$ equals

If $x = \sum_{n=0}^{\infty} a^{n}$,$y = \sum_{n=0}^{\infty} b^{n}$,$z = \sum_{n=0}^{\infty} c^{n}$,where $a, b, c$ are in $A.P.$ and $|a| < 1, |b| < 1, |c| < 1$,$abc \neq 0$,then:

If the altitudes of a triangle are in $A.P.$,then the sides of the triangle are in