Mathematics - Sequence & Series Question with Solution | TestHub

Let the G.P. be $2^a,2^{a+b},2^{a+2b},2^{a+3b},\dots$ with $a\ge 0,b\ge 1$ integers (increasing). ${\sum }_{n=1}^8{\log }_2{x}_n=8a+28b=196\Rightarrow 2a+7b=49$. $b$ must be odd and $a\ge 0:(a,b)=(0,7),(7,5),(14,3),(21,1)$. So there are 4 sequences. $x_4=2^{a+3b}$ is largest at $a=21,b=1:2^{24}$.