Exam 5: Induction and Recursion

In questions give a recursive definition with initial condition(s). -The function $f ( n ) = 5 n + 2 , n = 1,2,3 , \ldots$

Use mathematical induction to show that n lines in the plane passing through the same point divide the plane into $2 n \text { regions. }$

Use the Principle of Mathematical Induction to prove that $3 \mid \left( n ^ { 3 } + 3 n ^ { 2 } + 2 n \right) \text { for all } n \geq 1$

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

Consider the following program segment: i:=1 total :=1 while i$\text { Let } p \text { be the proposition "total } = \frac { i ( i + 1 ) } { 2 } \text { and } i \leq n . " \text { Use mathematical induction to prove that } p \text { is a loop }$ invariant.

Use the Principle of Mathematical Induction to prove that $2 \mid \left( n ^ { 2 } + n \right) \text { for all } n \geq 0$

Prove that all distributive law $A _ { 1 } \cap \left( A _ { 2 } \cup \cdots \cup A _ { n } \right) = \left( A _ { 1 } \cap A _ { 2 } \right) \cup \cdots \cup \left( A _ { 1 } \cap A _ { n } \right)$ is true for all $n \geq 3$

In questions give a recursive definition with initial condition(s) of the set S. - $\{ 3,7,11,15,19,23 , \ldots \}$

Verify that the program segment a:=2 b:=a+c is correct with respect to the initial assertion c = 3 and the final assertion b = 5.

A T -omino is a tile pictured at the right. Prove that every $2 ^ { n } \times 2 ^ { n } ( n > 1 )$ chessboard can be tiled with T-ominoes.

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.

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

In questions give a recursive definition with initial condition(s). -The set $\{ \ldots , - 4 , - 2,0,2,4,6 , \ldots \}$

Verify that the following program segment is correct with respect to the initial assertion T and the final assertion (x ≤ y ∧ max = y) ∨ (x > y ∧ max = x): if x ≤ y then max := y else max := x

In questions give a recursive definition with initial condition(s) of the set S. -All positive integer multiples of 5.

In questions give a recursive definition with initial condition(s). -The sequence $a _ { 1 } = 16 , a _ { 2 } = 13 , a _ { 3 } = 10 , a _ { 4 } = 7 , \ldots$

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

Prove that $\frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } = \frac { 2 ^ { n + 1 } - 2 - n } { 2 ^ { n } }$ for all $n \geq 1$

Find f(2) and f(3) if f(n)=f(n-1) / f(n-2), f(0)=2, f(1)=5 .

Suppose you wish to use the Principle of Mathematical Induction to prove that $1 \cdot 1 ! + 2 \cdot 2 ! + 3 \cdot 3 ! + \cdots + n \cdot n ! =$ $( n + 1 ) ! - 1 \quad \text { for all } n \geq 1 \text {. }$ (a) Write P(1) (b) Write P(5) (c) Write P(k) (d) Write P(k+1) (e) Use the Principle of Mathematical Induction to prove that P(n) is true for all $n \geq 1$