​ If , are roots of the equation x^2+ 5( √2)x +10 = 0, α > β and Pn = α^n – β^n for each positive integer n, then the value

Question

If \(\alpha\), \(\beta\) are roots of the equation x²+ 5( √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 ......... .

Answer 1

Correct answer 1

x² + 5(√2)x +10 = 0

& P_n = αⁿ - βⁿ (given)

Now put there values in above expression

= -\(\frac{10 P_{17} P_{18}}{-10P_{18} P_{17}} = 1\)