b)Prove that the following code returns the \(n^{th}\) odd natural number by structural induction.For instance , func(3)=5.Consider func(0) to be undefined.

At the given condition:

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Suppose n is an integer. Using the definitions of even and odd, prove that n is odd if and only if \(3n+1\) is even.

Let \(\displaystyle{F}_{{i}}\) be in the \(\displaystyle{i}^{{{t}{h}}}\) Fibonacc number, and let n be any positive integer \(\displaystyle\ge{3}\) Prove that \(\displaystyle{F}_{{n}}=\frac{1}{{4}}{\left({F}_{{{n}-{2}}}+{F}_{{n}}+{F}_{{{n}+{2}}}\right)}\)

Let R be the relation from X={1,2,3,5} to Y={0,3,4,9} defined by xRy if and only if \(\displaystyle{x}^{{2}}={y}\)

Consider the following pseudocode function. function Crunch\(\displaystyle{\left({x}\ {i}{s}\in{R}\right)}{\quad\text{if}\quad}{x}≥{100}\) then return x/100 else return \(x + Crunch(10 \cdot x)\) Compute Crunch(117).