Find the generating function of a_n an= ana t an-z (n23, 90=0, a,=l, az=z)%3D

Name: Find the generating function of a_n an= ana t an-z (n23, 90=0, a,=l, a
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Description: Find the generating function of a_n an= ana t an-z (n23, 90=0, a,=l, az=z)%3D

Answered: an= ant + an-z (n23, 90=0, a,=l, ag=2)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Find the generating function of a_n

an= ana t an-z (n23, 90=0, a,=l, az=z)
%3D

Expert Solution

Step 1

Given that $a_{0} = 0$ , $a_{1} = 1$ , $a_{2} = 2$ and $a_{n} = a_{n - 1} + a_{n - 2}$ .

Take $n = 3$ in the equation $a_{n} = a_{n - 1} + a_{n - 2}$ ,

$\begin{array}{rcl} a_{3} & = & 2 + 1 \\ = & 3 \end{array}$

Take $n = 4$ in the equation $a_{n} = a_{n - 1} + a_{n - 2}$ ,

$\begin{array}{rcl} a_{4} & = & a_{3} + a_{2} \\ = & 3 + 2 \\ = & 5 \end{array}$

For $n = 5$ ,

$\begin{array}{rcl} a_{5} & = & a_{4} + a_{3} \\ = & 5 + 3 \\ = & 8 \end{array}$

Hence the sequence is fibonacci sequence.

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