11.6: Exercises

Exercise $\PageIndex{1}$

Compute each of the terms $s_2,s_3,s_4,s_5,s_6$ for the sequence defined recursively by

$$\begin{equation*} s_n = \sqrt{s_{n-2}^2 + s_{n-1}^2}, \quad n \ge 2, \end{equation*}$$
with initial terms $s_0 = 3$ and $s_1 = 4\text{.}$

Solving by iteration.

In each of Exercises 2–8, use iteration to determine an expression for the $n^{th}$ term of the sequence as a formula in $n$ (and the initial term(s) of the sequence, if necessary).

In some of these, you may find the following formulas useful.

$$\begin{gather*} 1 + 2 + 3 + \dotsb + m = \frac{m (m+1)}{2} \\ 1^2 + 2^2 + 3^2 + \dotsb + m^2 = \frac{m (m+1) (2m+1)}{6} \\ r^0 + r^1 + r^2 + \dotsb + r^{m-1} = \frac{r^m - 1}{r - 1}, \quad r \ne 0,1 \end{gather*}$$

Exercise $\PageIndex{2}$

$a_n = 2na_{n-1}\text{,}$ $a_0 = 1 \text{.}$

Exercise $\PageIndex{3}$

$a_n = (2n-1)a_{n-1}\text{,}$ $a_0 = 1 \text{.}$

Exercise $\PageIndex{4}$

$a_n = a_{n-1} + 3^{n-1}\text{,}$ $a_0 = 1 \text{.}$

Exercise $\PageIndex{5}$

$a_n = a_{n-1} + n - 1\text{,}$ $a_0 = 1 \text{.}$

Exercise $\PageIndex{6}$

$a_n = a_{n-1} + n + n^2\text{,}$ $a_0 = 1 \text{.}$

Exercise $\PageIndex{7}$

$a_n = f(a_{n-1})\text{,}$ where $f(x)$ is the linear function $f(x) = mx + b$ for some fixed constants $m,b\text{,}$ and with arbitrary initial term $a_0\text{.}$

Exercise $\PageIndex{8}$

$a_n = 4a_{n-2}\text{,}$ $n \ge 2 \text{,}$ $a_0 = 1 \text{,}$ $a_1 = 2\text{.}$

Hint.

Treat the cases $n$ even and $n$ odd separately.

Exercise $\PageIndex{9}$

Fibonacci numbers are those that appear in the sequence defined recursively by

$$\begin{align*} a_n & = a_{n-1} + a_{n-2}, & n & \ge 2 \text{,} \end{align*}$$
for some choice of initial terms $a_0, a_1\text{.}$

See.

Example 11.2.3.

Using initial terms $a_0 = a_1 = 1\text{,}$ use mathematical induction to prove that every Fibonacci number $a_n$ satisfies $a_n \lt 2^n$ (except, of course, for $a_0\text{.}$

Example $\PageIndex{10}$

You are attempting to predict population dynamics on a yearly basis.

Suppose a population increases by a factor of $i$ each year. That is, if we set $p=100i\text{,}$ then the population increases by $p$ percent. (Careful: This is a description of the increase in population, not the total population. For example, $i = 1$ means that the population doubles.)

1. Write down a recurrence relation that expresses the population $P_n$ in the $n^{th}$ year relative to the previous year.
2. Use iteration to determine an expression for the population in the $n^{th}$ year as a formula in $n\text{,}$ $i\text{,}$ and the initial population $P_0\text{.}$
3. Suppose that on top of the natural population increase of $i$ percent per year, immigration increases the population by fixed amount $A$ people annually. Design a new recurrence relation for $P_n\text{,}$ and use iteration to determine an expression for the population in the $n^{th}$ year as a formula in $n\text{,}$ $i\text{,}$ $A\text{,}$ and the initial population $P_0\text{.}$

Example $\PageIndex{11}$

Explicitly describe how to construct the following logical statement in a finite number of steps using the inductive definition for $\mathscr{L}\text{,}$ the set of all possible logical statements, given in Example 11.4.1.

$$\begin{equation*} (p_1 \land p_2) \rightarrow ( (\neg p_3 \lor p_1) \Leftrightarrow (p_3 \land \neg p_2 ) ) \end{equation*}$$

Example $\PageIndex{12}$

The set $\mathscr{C}$ of constructible numbers can be defined inductively as follows.

Base clause.

Assume $1 \in \mathscr{C}\text{.}$

Inductive clauses.

Whenever $a,b \in \mathscr{C}\text{,}$ then so are

$$\begin{equation*} a+b, \quad ab, \quad a/b, \quad \sqrt{a} \text{.} \end{equation*}$$

Whenever $a,b \in \mathscr{C}$ with $a>b\text{,}$ then $a-b$ is also in $\mathscr{C}\text{.}$

Limiting clause.

The set $\mathscr{C}$ contains no elements other than those that can be obtained through a finite number of applications of the base and/or inductive clauses.

Explicitly verify, by listing each application of the relevant clauses, that the roots of the polynomial $2x^2 - 3x + \frac{7}{8}$ are both constructible numbers.

Example $\PageIndex{13}$

Consider the following inductively defined set $A \subseteq \mathbb{N}\text{.}$

Base clause.
Assume $32879 \in A\text{.}$

Inductive clauses.
When $a$ is an element of $A\text{,}$ then each of the prime factors of $a$ is also an element of $A\text{.}$

Whenever prime $p$ is an element of $A\text{,}$ then $p+1$ is also an element of $A\text{.}$

Limiting clause.
The set $A$ contains no elements other than those that can be obtained through a finite number of applications of the base and/or inductive clauses.

Determine all elements of $A\text{.}$

Hint.

To help with this question, you may wish to search for “list of small primes” on the internet.

Example $\PageIndex{14}$

Devise an algorithm that will produce an answer to the following question in a finite number of applications of the inductive clause that we used to define the natural numbers in Example 11.4.2.

Given $m,n \in \mathbb{N}$ with $m \ne n\text{,}$ is $m \gt n$ or is $n \gt m$ ?