Mathematics - Sequence & Series Question with Solution | TestHub

$\frac{\frac{n}{2}(2+(n-1)2)}{\frac{n}{2}(2\times 4+(n-1)3)}=\frac{2n}{3n+5}=\frac{20}{7{\log }_{10}x}$. $n={\log }_{10}x\left[1+\frac{1}{2}+\frac{1}{4}+\dots \right]=2{\log }_{10}x$, so ${\log }_{10}x=\frac{n}{2}$. Then $\frac{2n}{3n+5}=\frac{40}{7n}\Rightarrow 7n^2=60n+100\Rightarrow 7n^2-60n-100=0\Rightarrow (n-10)(7n+10)=0$. $n=10$ (the root $-\frac{10}{7}$ is rejected, since $n$ counts terms). So ${\log }_{10}x=5$ and $x=10^5$.