Exam 5: A: Induction and Recursion

Verify that the following program segment is correct with respect to the initial assertion T and the final $\text { assertion } ( x \leq y \wedge \max = y ) \vee ( x > y \wedge \max = x ) :$

Let S be the set of positive integers defined by: Basis step: 4 $\in$ S . Recursive step: If n $\in$ S , then $5 n + 2 \in S$ and $5 n + 2 \in S$ (a) Show that if $n \in S$ , then $n \equiv 4$ (mod 6). (b) Show that there exists an integer $m \equiv 4$ (mod 6) that does not belong to $S$

give a recursive definition (with initial condition(s)) of $\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots )$ - $a _ { n } = \sqrt { 2 }$

give a recursive definition with initial condition(s). -The set {1, 5, 9, 13, 17, . . .}

give a recursive definition with initial condition(s). -The set {0, 3, 6, 9, . . .}

Let $a _ { 1 } = 2 , a _ { 2 } = 9 \text {, and } a _ { n } = 2 a _ { n - 1 } + 3 a _ { n - 2 } \text { for } n \geq 3 \text {. Show that } a _ { n } \leq 3 ^ { n } \text { for all positive integers } n \text {. }$

give a recursive definition (with initial condition(s)) of $\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots )$ - $a _ { n } = 2 ^ { n }$

Use mathematical induction to prove that $1 + 2 ^ { n } \leq 3 ^ { n } \text { for all } n \geq 1$

Use mathematical induction to prove that every integer amount of postage of six cents or more can be formed using 3-cent and 4-cent stamps.

give a recursive definition (with initial condition(s)) of $\left\{ a _ { n } \right\} \quad ( n = 1,2,3 , \ldots )$ - $a _ { n } = ( n + 1 ) / 3$

Use mathematical induction to prove that $2 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 0 \text {. }$