If $\alpha, \beta$ are roots of the equation $x^{2}+5(\sqrt{2}) x+10=0$, $\alpha\beta$ and $P_{n}=\

Question

If $\alpha, \beta$ are roots of the equation $x^{2}+5(\sqrt{2}) x+10=0$, $\alpha>\beta$ and $P_{n}=\alpha^{n}-\beta^{n}$ for each positive integer $n$, then the value of $\left(\frac{P_{17} P_{20}+5 \sqrt{2} P_{17} P_{19}}{P_{18} P_{19}+5 \sqrt{2} P_{18}^{2}}\right)$ is equal to

Professor PC Thomas Classes · Answer

(1) : $\alpha, \beta$ are roots of the equation $x^{2}+5(\sqrt{2}) x+10=0, \alpha>\beta$\ $P_{n}=\alpha^{n}-\beta^{n}$, where $n$ is a positive integer.\ Now, $\frac{P_{17}\left[P_{20}+5 \sqrt{2} P_{19}ight]}{P_{18}\left[P_{19}+5 \sqrt{2} P_{18}ight]}$ $$ =\frac{P_{17}}{P_{18}}\left[\frac{\left(\alpha^{20}-\beta^{20}ight)+5 \sqrt{2}\left(\alpha^{19}-\beta^{19}ight)}{\left(\alpha^{19}-\beta^{19}ight)+5 \sqrt{2}\left(\alpha^{18}-\beta^{18}ight)}ight] $$ $=\frac{P_{17}}{P_{18}}\left[\frac{10 \beta^{18}-10 \alpha^{18}}{10 \beta^{17}-10 \alpha^{17}}ight]=\frac{P_{17}}{P_{18}} \frac{\left(-10 \cdot P_{18}ight)}{\left(-10 \cdot P_{17}ight)}=1$\