Question 1

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 $: = A [ 1 ] \cdot A [ 2 ] \cdots A [ n + 1 ] "$

Accepted Answer

The answer of The following while loop is annotated with...

Question 2

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.

Accepted Answer

The answer of A sequence \(a _ { 0 }...

Question 3

$\text { Compute } \sum _ { k = 0 } ^ { 3 } \frac { 1 } { 2 ^ { k } }$

Accepted Answer

The answer of \(\text { Compute } \sum _ {...

Question 4

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.

Accepted Answer

The answer of A single pair of rabbits (male and...

Question 5

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.

Accepted Answer

The answer of A sequence is deﬁned recursively as follows:...

Question 6

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

Accepted Answer

The answer of Use repeated division by 2 to ﬁnd...

Question 7

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

Accepted Answer

The answer of Use a summation symbol to rewrite the...

Question 8

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

Accepted Answer

The answer of Transform the following summation by making the...

Question 9

$\text { Compute } \sum _ { k = 1 } ^ { 4 } k ^ { 2 } \text {. }$

Accepted Answer

The answer of \(\text { Compute } \sum _ {...

Question 10

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
$$\sum _ { k = 1 } ^ { n } \left( 2 a _ { k } - 3 b _ { k } \right) = 2 \sum _ { k = 1 } ^ { n } a _ { k } - 3 \sum _ { k = 1 } ^ { n } b _ { k } .$$

Accepted Answer

The answer of Use the recursive definition of summation together...

Question 11

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.

Accepted Answer

The answer of Define a set $S$ recursively as follows:
I....

Question 12

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.

Accepted Answer

The answer of Suppose a certain amount of money is...

Question 13

Use iteration to find an explicit formula for the sequence $b _ { 0 } , b _ { 1 } , b _ { 2 } , \ldots$ defined recursively as follows:
$$\begin{array} { l } 
b _ { k } = 2 b _ { k - 1 } + 3 \quad \text { for all integers } k \geq 1 \
b _ { 0 } = 1 .
\end{array}$$ 
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 $r \neq 1$.

Accepted Answer

The answer of Use iteration to find an explicit formula...

Question 14

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.

Accepted Answer

The answer of In a Double Tower of Hanoi with...

Question 15

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.

Accepted Answer

The answer of Use the formula \(1 + r +...

Question 16

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.

Accepted Answer

The answer of In a Triple Tower of Hanoi, there...

Question 17

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.

Accepted Answer

The answer of Use strong mathematical induction to prove that...

Exam 5: Sequences, Mathematical Induction, and Recursion

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.

$\text { Compute } \sum _ { k = 0 } ^ { 3 } \frac { 1 } { 2 ^ { k } }$

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: $\sum _ { k = 0 } ^ { n } \frac { k ^ { 2 } } { k + n }$

$\text { Compute } \sum _ { k = 1 } ^ { 4 } k ^ { 2 } \text {. }$

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.

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.

Exam 5: Sequences, Mathematical Induction, and Recursion

Compute ∑k=0312k\text { Compute } \sum _ { k = 0 } ^ { 3 } \frac { 1 } { 2 ^ { k } } Compute ∑k=03​2k1​

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: 1−12+13−14+15−161 - \frac { 1 } { 2 } + \frac { 1 } { 3 } - \frac { 1 } { 4 } + \frac { 1 } { 5 } - \frac { 1 } { 6 }1−21​+31​−41​+51​−61​

Transform the following summation by making the change of variable i = k + 1: ∑k=0nk2k+n\sum _ { k = 0 } ^ { n } \frac { k ^ { 2 } } { k + n }∑k=0n​k+nk2​

Compute ∑k=14k2. \text { Compute } \sum _ { k = 1 } ^ { 4 } k ^ { 2 } \text {. } Compute ∑k=14​k2.

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

$\text { Compute } \sum _ { k = 0 } ^ { 3 } \frac { 1 } { 2 ^ { k } }$

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: $\sum _ { k = 0 } ^ { n } \frac { k ^ { 2 } } { k + n }$

$\text { Compute } \sum _ { k = 1 } ^ { 4 } k ^ { 2 } \text {. }$

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.