I.

a$)$ Prove the following using induction, for all $n$ Natural number. show all the steps

$\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+$cdots$+\frac{1}{2^n-1}+\frac{1}{2^n}1+\frac{n}{2}$

b$)$ Convert the following to symbolic form then using the rules of inferences show if the argument is valid or invalid

"All Ferrari owners are speeders. No owners of SUV buy premium gas. Car owners that do not buy premium gas never speed. So Ferarri owners do not own SUV'.

II$.$

For $=\{a,b\}$ give a DFA for all strings not containing aa substring.