11.6: Exercises
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*}
\(a_n = 2na_{n-1}\text{,}\) \(a_0 = 1 \text{.}\)
\(a_n = (2n-1)a_{n-1}\text{,}\) \(a_0 = 1 \text{.}\)
\(a_n = a_{n-1} + 3^{n-1}\text{,}\) \(a_0 = 1 \text{.}\)
\(a_n = a_{n-1} + n - 1\text{,}\) \(a_0 = 1 \text{.}\)
\(a_n = a_{n-1} + n + n^2\text{,}\) \(a_0 = 1 \text{.}\)
\(a_n = f(a_{n-1})\text{,}\) where \(f(x)\) is the linear function \(f(x) = mx + b \) for some fixed constants \(m,b\text{,}\) and with arbitrary initial term \(a_0\text{.}\)
\(a_n = 4a_{n-2}\text{,}\) \(n \ge 2 \text{,}\) \(a_0 = 1 \text{,}\) \(a_1 = 2\text{.}\)
- Hint.
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Treat the cases \(n\) even and \(n\) odd separately.
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.
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 \(a_0\text{.}\)
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.)
- Write down a recurrence relation that expresses the population \(P_n\) in the \(n^{th}\) year relative to the previous year.
- 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 \(P_0\text{.}\)
- 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 \(P_0\text{.}\)
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*}
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.
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.
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To help with this question, you may wish to search for “list of small primes” on the internet.
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 \(n \gt m\) ?