Explanation

(a)
If $a,\ b,c\$ are in GP, then $b^{2} = ac$
Taking log on both sides, we get
$2\log_{e}{b = \log_{e}{a + \log_{e}c}}$
$\Rightarrow 2n\log_{e}{b = n\log_{e}{a + n\log_{e}c}}$
$\Rightarrow 2\log_{e}{b^{n} = \log_{e}{a^{n} + \log_{e}c^{n}}}$
$\Rightarrow \log_{e}{a^{n},\log_{e}{b^{n},}}\log_{e}c^{n}$ be in AP.