Exam 5: Sequences, Mathematical Induction, and Recursion

The following while loop is annotated with a pre- and post-condition and also a loop invariant. Use the loop invariant theorem to prove the correctness of the loop with respect to the pre- and post-conditions. $[$ Pre-condition: product $= A [ 1 ]$ and $i = 1 ]$ while $( i \neq m )$ 1. $i : = i + 1$ 2. product := product. $A [ i ]$ end while $[$ Post-condition: product $= A [ 1 ] \cdot A [ 2 ] \cdots A [ m ] ]$ loop invariant: $I ( n )$ is $" i = n + 1$ and product $: = A [ 1 ] \cdot A [ 2 ] \cdots A [ n + 1 ] "$

A sequence $a _ { 0 } , a _ { 1 } , a _ { 2 } , \ldots$ satisfies the recurrence relation $a _ { k } = 4 a _ { k - 1 } - 3 a _ { k - 2 }$ with initial conditions $a _ { 0 } = 1$ and $a _ { 1 } = 2$ . Find an explicit formula for the sequence.

A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions: (a) Rabbit pairs are not fertile during their first two months of life, but thereafter they give birth to four new male/female pairs at the end of every month; (b) No deaths occur. Let $s _ { n } =$ the number of pairs of rabbits alive at the end of month $n$ , for each integer $n \geq 1$ , and let $s _ { 0 } = 1$ . Find a recurrence relation for $s _ { 0 } , s _ { 1 } , s _ { 2 } , \ldots$ Justify your answer carefully.

A sequence is defined recursively as follows: $a _ { 0 } = 2 \quad \text { and } \quad a _ { k } = 4 a _ { k - 1 } + 1 \quad \text { for all } k \geq 1 .$ It is proposed that an explicit formula for this sequence is $a _ { n } = \frac { 7 \cdot 4 ^ { n } - 1 } { 3 }$ Use mathematical induction to check whether this proposed formula is correct.

Use repeated division by 2 to find the binary representation of the number 1032. Show your work.

Use a summation symbol to rewrite the following: $1 - \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 4 } + \frac { 1 } { 5 } - \frac { 1 } { 6 }$

Transform the following summation by making the change of variable i = k + 1: $\sum _ { k = 0 } ^ { n } \frac { k ^ { 2 } } { k + n }$

Use the recursive definition of summation together with mathematical induction to prove that for all positive integers $n$ , if $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$ and $b _ { 1 } , b _ { 2 } , \ldots , b _ { n }$ are real numbers, then $\sum _ { k = 1 } ^ { n } \left( 2 a _ { k } - 3 b _ { k } \right) = 2 \sum _ { k = 1 } ^ { n } a _ { k } - 3 \sum _ { k = 1 } ^ { n } b _ { k } .$

Define a set $S$ recursively as follows: I. BASIS: $11 \in S$ II. RECURSION: a. If $s \in S$ , then $0 s \in S$ and $s 0 \in S$ b. If $x$ is any string (including the null string) such that $1 x 1 \in S$ , then $10 x 1 \in S$ and $1 x 01 \in S$ III. RESTRICTION: No strings other than those derived from I and II are in $S$ . a. Is $010010 \in S$ ? Justify your answer. b. Is $011011 \in S$ ? Justify your answer.

Suppose a certain amount of money is deposited into an account paying $4 \%$ annual interest, compounded quarterly (i.e., four times a year). For each positive integer $n$ , let $S _ { n } =$ the amount on deposit at the end of the $n$ th quarter, and let $S _ { 0 }$ be the initial amount deposited. (a) Find a recurrence relation for $S _ { 0 } , S _ { 1 } , S _ { 2 } , \ldots$ , assuming no additional deposits or withdrawals for a 3-year period. (b) If $S _ { 0 } = \$ 5000$ , find the amount of money on deposit at the end of three years. (c) Find the APR for the account.

Use iteration to find an explicit formula for the sequence $b _ { 0 } , b _ { 1 } , b _ { 2 } , \ldots$ defined recursively as follows: =2+3 for all integers k\geq1 =1. If appropriate, simplify your answer using one of the following reference formulas: (a) $1 + 2 + 3 + \cdots + n = \frac { n ( n + 1 ) } { 2 }$ for all integers $n \geq 1$ . (b) $1 + r + r ^ { 2 } + \cdots + r ^ { m } = \frac { r ^ { m + 1 } - 1 } { r - 1 }$ for all integers $m \geq 0$ and all real numbers $r \neq 1$ .

In a Double Tower of Hanoi with Adjacency Requirement there are three poles in a row and 2n disks, two of each of n different sizes, where n is any positive integer. Initially pole A (at one end of the row) contains all the disks, placed on top of each other in pairs of decreasing size. Disks may only be transferred one-by-one from one pole to an adjacent pole and at no time may a larger disk be placed on top of a smaller one. However a disk may be placed on top of another one of the same size. Let C be the pole at the other end of the row and let $s _ { n } = \left[ \begin{array} { l } \text { the minimum number of moves } \\\text { needed to transfer a tower of } 2 n \\\text { disks from pole } A \text { to pole } C\end{array} \right]$ (a) Find $s _ { 1 }$ and $s _ { 2 }$ . (b) Find a recurrence relation expressing $s _ { k }$ in terms of $s _ { k - 1 }$ for all integers $k \geq 2$ . Justify your answer carefully.

Use the formula $1 + r + r ^ { 2 } + \cdots + r ^ { n } = \frac { r ^ { n + 1 } - 1 } { r - 1 }$ (for all real numbers $r \neq 1$ and for all integers $n \geq 0$ ) to find $2 + 2 ^ { 2 } + 2 ^ { 3 } + \cdots + 2 ^ { m }$ where m is an integer that is at least 1.

In a Triple Tower of Hanoi, there are three poles in a row and 3n disks, three of each of n different sizes, where n is any positive integer. Initially, one of the poles contains all the disks placed on top of each other in triples of decreasing size. Disks are transferred one by one from one pole to another, but at no time may a larger disk be placed on top of a smaller disk. However, a disk may be placed on top of one of the same size. Let $t _ { n}$ be the minimum number of moves needed to transfer a tower of 3n disks from one pole to another. Find a recurrence relation for $t _ { 1 } , t _ { 2 } , t _ { 3 } , \ldots$ Justify your answer carefully.

Use strong mathematical induction to prove that for all integers $n \geq 2$ either n is prime or n is a product of prime numbers.