If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α > β and P n = α n - β n for each positive integer n , then the value of P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 is equal to

Question

If &#945; , &#946; are roots of the equation x 2 + 5 2 x + 10 = 0 , &#945; &#62; &#946; and P n = &#945; n - &#946; n for each positive integer n , then the value of P 17 P 20 + 5 2 P 17 P 19 P 18 P 19 + 5 2 P 18 2 is equal to

Aicharya · Accepted Answer

Given, $x^{2}+5\sqrt{2}x+10=0$ and $P_{n}=\alpha^{n}-\beta^{n}$ Now $\frac{P_{17}P_{20}+5\sqrt{2}P_{17}P_{19}}{P_{18}P_{19}+5\sqrt{2}P_{18}^{2}}=\frac{P_{17}(P_{20}+5\sqrt{2}P_{19})}{P_{18}(P_{19}+5\sqrt{2}P_{18})}$ $=\frac{P_{17}(\alpha^{20}-\beta^{20}+5\sqrt{2}(\alpha^{19}-\beta^{19}))}{P_{18}(\alpha^{19}-\beta^{19}+5\sqrt{2}(\alpha^{18}-\beta^{18}))}$ $=\frac{P_{17}(\alpha^{19}(\alpha+5\sqrt{2})-\beta^{19}(\beta+5\sqrt{2}))}{P_{18}(\alpha^{18}(\alpha+5\sqrt{2})-\beta^{18}(\beta+5\sqrt{2}))}$ Since $\alpha+5\sqrt{2}=-\frac{10}{\alpha}$ and $\beta+5\sqrt{2}=-\frac{10}{\beta}$

Now put these values in above expression

$\frac{P_{17} (α^{19} (α + 5 \sqrt{2}) - β^{19} (β + 5 \sqrt{2}))}{P_{18} (α^{18} (α + 5 \sqrt{2}) - β^{18} (β + 5 \sqrt{2}))} = \frac{- 10 P_{17} P_{18}}{- 10 P_{18} P_{17}} = 1$

If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α > β and P n = α n -…

Solution

If &#945; , &#946; are roots of the equation x 2 + 5 2 x + 10 = 0 , &#945; &#62; &#946; and P n = &#945; n -…

Solution

If α , β are roots of the equation x 2 + 5 2 x + 10 = 0 , α > β and P n = α n -…