Question-105536

Question Number 105536 by I want to learn more last updated on 29/Jul/20

Answered by adhigenz last updated on 29/Jul/20

α + β = 5, α β = - 2

x_{n} = α^{n} - β^{n}

= (α + β) (α^{n - 1} - β^{n - 1}) - α β (α^{n - 2} - β^{n - 2})

= 5 x_{n - 1} + 2 x_{n - 2}

Substitute for n = 2002 :

x_{2002} = 5 x_{2001} + 2 x_{2000}

\Rightarrow 2 x_{2000} - x_{2002} = - 5 x_{2001}

\frac{2 x_{2000} - x_{2002}}{x_{2001}} = \frac{- 5 x_{2001}}{x_{2001}} = - 5

Commented by I want to learn more last updated on 29/Jul/20

Wow, thanks sir, but how do we know where to stop in the expansion of

α^{n} - β^{n} and how we get the expansion .

Commented by adhigenz last updated on 30/Jul/20

We can relate on what is on the given

question like α + β and α β then find the

relationship involving x_{n} = α^{n} - β^{n}

α^{n} - β^{n} = (α + β) (α^{n - 1} - β^{n - 1}) + α β^{n - 1} - α^{n - 1} β

= (α + β) (α^{n - 1} - β^{n - 1}) - α β (α^{n - 2} - β^{n - 2})

Commented by I want to learn more last updated on 30/Jul/20

I appreciate sir