DRDO CSE 2022 Paper 1 | Question: 15

Question

What is the generating function corresponding to Fibonacci series.

\[F_{n}=F_{n-1}+F_{n-2} .\]

Note that $F_{0}=F_{1}=1$.

Answer 1

The fibonacci sequence can be wriiten as per definition given :

$a_{0},$$a_{1},a_{2},a_{3},a_{4},a_{5},a_{6},a_{7},....$ = $1,1,2,3,5,8,13,21,…...$

Therefor the ordinary generating function corresponding to the given sequence =

$G(x) = 1.x^{0} + 1.x^{1} +2.x^{2}+3.x^{3} +5.x^{4}+8.x^{5}....$        $\textbf{......(1) }$

            $= 1.+ 1.x^{1} +2.x^{2}+3.x^{3} +5.x^{4}+8.x^{5}....$

$xG(x) = 1.x^{1} + 1.x^{2} +2.x^{3}+3.x^{4} +5.x^{5}+....$     $\textbf{......(2) }$

Subtracting $\textbf{(2) }$ from $\textbf{(1) }$

$G(x) - xG(x) = 1+ 0x^{1} + 1x^{2} +1x^{3} + 2x^{4} + 3x^{5} + 5x^{6}+.....$

$= 1+ 0x^{1} + x^{2}(1x^{0} + 1x^{1} + 2x^{2} + 3x^{3}+.....)$

$= 1+ 0x^{1} + x^{2}G(x)$

$\Rightarrow G(x) = \frac{1}{1-x-x^{2}}$

Comments

would it be fibonacci series or fibonacci  sequence ?

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@ankitgupta.1729 sir

correction done