Exam 5: Induction and Recursion

In questions give a recursive definition with initial condition(s) of the set S. - $\{ \ldots , - 5 , - 3 , - 1,1,3,5 , \ldots \}$

Suppose you wish to prove that the following is true for all positive integers n by using the Principle of Mathematical Induction: 1+3+5+...+(2 n-1)=n² . (a) Write P(1) (b) Write P(72) (c) Write P(73) (d) Use P(72) to prove P(73) (e) Write P(k) (f) Write P(k+1) (g) Use the Principle of Mathematical Induction to prove that P(n) is true for all positive integers n

In questions give a recursive definition with initial condition(s). -The function $f ( n ) = n ! , n = 0,1,2 , \ldots$

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

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

Use the Principle of Mathematical Induction to prove that any integer amount of postage from 18 cents on up can be made from an infinite supply of 4-cent and 7-cent stamps.

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 $\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots )$ - $a _ { n } = ( n + 1 ) / 3$

Prove that $\sum _ { j = n } ^ { 2 n - 1 } ( 2 j + 1 ) = 3 n ^ { 2 } \text { for all positive integers } n$

give a recursive definition (with initial condition(s)) of $\left\{ a _ { n } \right\} ( n = 1,2,3 , \ldots )$ - $a _ { n } = 3 n - 5$

In questions give a recursive definition with initial condition(s). -The set $\{ 1,1 / 3,1 / 9,1 / 27 , \ldots \}$

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

In questions give a recursive definition with initial condition(s). -The Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, . . . .