If-a-n-3-a-n-1-a-n-1-and-a-0-1-a-2-2-find-a-n-

Question Number 93173 by I want to learn more last updated on 11/May/20

If a_{n + 3} = \frac{a_{n - 1}}{a_{n + 1}}, and a_{0} = 1, a_{2} = 2

find a_{n}

Answered by prakash jain last updated on 12/May/20

a_{n + 3} = \frac{a_{n - 1}}{a_{n + 1}}

b_{n} = \ln a_{n}

\ln a_{n + 3} = \ln a_{n - 1} - \ln a_{n + 1}

b_{n + 3} = b_{n - 1} - b_{n + 1}

b_{n + 3} + b_{n + 1} - b_{n - 1} = 0

Characteristic equation

x^{4} + x^{2} - 1 = 0

x^{2} = \frac{- 1 \pm \sqrt{5}}{2}

x = \pm \sqrt{\frac{- 1 \pm \sqrt{5}}{2}}

x = \pm \sqrt{\frac{\sqrt{5} - 1}{2}} and \pm i \sqrt{\frac{\sqrt{5} + 1}{2}}

b_{n} = c_{1} {(\sqrt{\frac{\sqrt{5 - 1}}{2}})}^{n} + c_{2} {(- \sqrt{\frac{\sqrt{5} - 1}{2}})}^{n}

+ c_{3} {(i \sqrt{\frac{\sqrt{5} + 1}{2}})}^{n} + c_{4} {(- i \sqrt{\frac{\sqrt{5} + 1}{2}})}^{n}

b_{n} = \frac{1}{2^{n / 2}} {(c_{1} + {(- 1)}^{n} c_{2}) {(\sqrt{5} - 1)}^{n / 2} + i (c_{3} + {(- 1)}^{n} c_{4}) {(\sqrt{5} + 1)}^{n / 2}

a_{n} = \exp (\frac{1}{2^{n / 2}} {(c_{1} + {(- 1)}^{n} c_{2}) {(\sqrt{5} - 1)}^{n / 2} + i (c_{3} + {(- 1)}^{n} c_{4}) {(\sqrt{5} + 1)}^{n / 2})

where c_{1,} c_{2}, c_{3} and c_{4} are arbitarary

constants .

You have only given two initial

conditions . You need 4 to

determine c_{1} to c_{4}

Commented by prakash jain last updated on 12/May/20

e x p (x) = e^{x}

Commented by I want to learn more last updated on 12/May/20

Thanks sir, i appreciate