11.6: Exercises
( \newcommand{\kernel}{\mathrm{null}\,}\)
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 ?