Question

Let ' $X$ ' denotes the value of the product
$\left(1+a+a^2+a^3+\ldots \ldots \infty\right)\left(1+b+b^2+b^3+\ldots \ldots \infty\right)$
where ${ }^{\prime} a$ ' and ' $b$ ' are the roots of the quadratic equation $11 x ^2-4 x -2=0$ and ' $Y$ ' denotes the numerical value of the infinite series
$\left(\log _c 2\right)^0\left(\log _c 5^{4^0}\right)+\left(\log _c 2\right)^1\left(\log _c 5^{4^1}\right)+\left(\log _c 2\right)^2\left(\log _c 5^{4^2}\right)+\left(\log _c 2\right)^3\left(\log _c 5^{4^3}\right)+\ldots \ldots \infty$
where $c=2000$. If the value of $(X Y)$ can be expressed as rational $\frac{p}{q}$ (where $\left.p, q \in N\right)$, find $(p+q)$.

Answer 1

$ X =\frac{1}{1- a } \cdot \frac{1}{1- b }=\frac{1}{1-( a + b )+ ab } $ where $a + b =\frac{4}{11}$ and $ab =-\frac{2}{11}$
$\therefore X=\left(1-\frac{4}{11}-\frac{2}{11}\right)^{-1}=\left(1-\frac{6}{11}\right)^{-1} \Rightarrow X=\frac{11}{5} $

Roots are
$\alpha, \beta =\frac{4 \pm \sqrt{16+88}}{22}$
$ =\frac{4 \pm \sqrt{104}}{22}=\frac{2 \pm \sqrt{26}}{11}$

$\text { for } Y: a=\log _b 5 ; r=\frac{4 \log _b 5 \cdot \log _b 2}{\log _b 5}=4 \log _b 2$
$\therefore Y =\frac{\log _{ b } 5}{1-4 \log _{ b } 2}=\frac{\log _{ b } 5}{\log _{ b } b -\log _{ b } 16}=\frac{\log _{ b } 5}{\log _{ b } \frac{ b }{16}}=\frac{\log _{2000} 5}{\log _{2000} 125}=\log _{125} 5=\frac{1}{3} \Rightarrow Y =\frac{1}{3}$
$\Rightarrow XY =\frac{11}{5} \cdot \frac{1}{3}=\frac{11}{15}$