If x,yR, x0, y = _10x+_10x^1/3+_10x^1/9+ up to terms, and 2+4+6++2y3+6+9++3y = 4_10x, the…

Question

If $x,y\in\mathbb{R}$, $x>0$, $y = \log_{10}x+\log_{10}x^{1/3}+\log_{10}x^{1/9}+\cdots$ up to $\infty$ terms, and $\dfrac{2+4+6+\cdots+2y}{3+6+9+\cdots+3y} = \dfrac{4}{\log_{10}x}$, then the ordered pair $(x,y)$ is equal to:

Accepted Answer

$\dfrac{2(1+2+\cdots+y)}{3(1+2+\cdots+y)} = \dfrac23 = \dfrac{4}{\log_{10}x} \Rightarrow \log_{10}x=6 \Rightarrow x=10^6.$

$$y = \log_{10}x\left(1+\frac13+\frac19+\cdots\right) = \log_{10}x\cdot\frac{1}{1-1/3} = 6\times\frac32 = 9.$$

$(x,y) = (10^6, 9).$

Solutions · 1