Question-162825

Question Number 162825 by KONE last updated on 01/Jan/22

Answered by mr W last updated on 01/Jan/22

2 .

P_{n + 2} - 2 {X P}_{n + 1} + P_{n} = 0

t^{2} - 2 X t + 1 = 0 (c h a r . e q n .)

t = X \pm \sqrt{X^{2} - 1}

P_{n} = A {(X + \sqrt{X^{2} - 1})}^{n} + B {(X - \sqrt{X^{2} - 1})}^{n}

P_{0} = A + B = 1

P_{1} = A (X + \sqrt{X^{2} - 1}) + B (X - \sqrt{X^{2} - 1}) = X

\Rightarrow A = B = \frac{1}{2}

P_{n} = \frac{{(X + \sqrt{X^{2} - 1})}^{n} + {(X - \sqrt{X^{2} - 1})}^{n}}{2}

P_{n} = \frac{{(X + \sqrt{X^{2} - 1})}^{n} + {(X + \sqrt{X^{2} - 1})}^{- n}}{2}

P_{n} = \frac{e^{n \ln (X + \sqrt{X^{2} - 1})} + e^{- n \ln (X + \sqrt{X^{2} - 1})}}{2}

P_{n} = \cosh [n \ln (X + \sqrt{X^{2} - 1})]

P_{n} = \cosh (n \cosh^{- 1} X)

Commented by KONE last updated on 02/Jan/22

m e r c i a v o u s

Commented by Tawa11 last updated on 02/Jan/22

Great sir

Answered by KONE last updated on 01/Jan/22

b e s o i n d^{'} a i d e

Facebook Tweet Pin