Question $1(10$ points$)$: Prove the following mathematical statement using mathematical induction, clearly identify each step in the process, and show all algebraic steps$\mathrm{.}2$i $=($n $-1).2$n$+1+2$Question $2(10$ points$)$ Prove the following mathematical statement using mathematical induction, show ALL the steps in the process.an$+1-1$Question $3(10$ points$)$ Solve the recurrence relation below using the iteration method. Clearly state any reasonable assumptions you make. Find an exact formula of T$($n$),$ and then give an asymptotic expression $($i$.$e$.$ the big$-$o expression$).$T $($n$)=1($n$)=27(2+$ lognQuestion $4(20$ points$)$: Solve the following recurrence using the a$)$ the substitution method and b$)$ using the iteration method. In the case of the iteration method give an exact formula for T$($n$)$ and then provide an asymptotic expression for T$($n$)($i$.$e$.$ a big$-$O expression$).$ Show ALL steps in the process.T$($n$)=1$T $($n$)=7$T $($G$)+$ n$3,$ vn $2$Question $5(10$ points$)$ Solve the recurrence relation below using the iteration method, make any reasonable assumptions. Find an exact formula of T$($n$),$ and then give an asymptotic expression $($i$.$e$.$ the big$-$O expression$).$T $(1)=1$T $($n$)=$ T$+2$n