Exercises
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Exercises
Find the next four terms of the following recursively defined sequences.
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The following questions refer to the Fibonacci sequence defined by $F_k=F_{k-1}+F_{k-2}$.
- Explain why this is equivalent to $F_{k+1}=F_k + F_{k-1}$.
- Prove that $F_k=3F_{k-3}+2F_{k-4}$, for $k\ge 4$.
- Prove that $F_k^2-F_{k-1}^2 = F_k F_{k+1}-F_{k-1}F_{k+1}$.
- Show by mathematical induction that $F_{n+2}F_n - F_{n+1}^2 = F_{k-1}F_{k+2}$ for all integers $k\ge 1$.
- Show that for $F_n \ge 0$, $F_n=\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$.
Infer an explicit formula for the following sequences by looking at the first several terms.
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In the lectures we saw that if $a_n=k a_{n-1}$ then $a_n = a_0 k^n$. Prove this using mathematical induction.- Suppose a population increases by 3.5% per year. Find a formula that represents the size of the population after $n$ years.
- The concentration of a medication in the patient's blood has to stay above 100 mg. An initial dosage has 500 mg. If the concentration in the blood stream reduces by 15% every 6 hours develop a model that describes the concentration after n 6 hour intervals.
- Using your model from the previous question, write out a numeric solution and use it to determine the time it takes for the medication to drop to the minimum safe level.
- Suppose you owe \$1,200 on a credit card where you pay 11% interest. If you make the minimum \$100 per month, develop a numerical solution that represents the value of the account after n months. How many payments will you have to make to pay off the account?
- How does the solution to the previous question change if you make $50 in charges every month?
Explorations
- Give a recursive definition for $\Pi_{i=0}^n a_i$.
- Show that $\Pi_{i=0}^n (a_i b_i) = \Pi_{i=0}^n a_i \Pi_{i=0}^n b_i$ where $a_i, b_i \in \mathbb{R}$.
- Show that $\Pi_{i=1}^n (c a_i) = c^n \Pi_{i=1}^n a_i$ where $a_i, c \in \mathbb{R}$.
- An annuity is an investment where you make a single initial payment then the account earns a fixed interest rate every month ($r$) and you get a fixed payment from the account every month ($b$). Develop a difference equation that represents the amount remaining in the account after $n$ months. (This same model can be used to describe the remaining balance on a mortgage payment or credit card.)
- To save for their retirement, John puts a fixed amount every month ($R$) into an account that earns a fixed monthly interest rate ($r$). Develop both a difference equation and an explicit formula for the value of the account after $n$ months. How could you modify the formula to represent the value after $t$ years still assuming that the payments and interest are made monthly?
- Prove the following: Suppose $x(n)$ is a solution to the equation $x_n = ax_{n-1}$ and $y(n)$ is a solution to the equation $y_n=ay_{n-1} + b(n)$ then $z(n) = x(n) + y(n) = x_n + y_n$ is also a solution to $y_n=ay_{n-1} + b(n)$. (Note that the only difference between $x_n$ and $y_n$ is that $y_n$ has an additional term(s) that only depend on $n$. This theorem says that if you can find one solution to the general equation $y_n$ then you can get a more general solution by adding the solution to $x_n=ax_{n-1}$ to it.)
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Find the next four terms of the sequence given by $a_n=3a_{n-1}+1$, $a_0=1$
$$a_0 = 1$$ $$a_1=3 \cdot 1 + 1 = 4$$ $$a_2=3 \cdot 4 + 1 = 13$$ $$a_3=3 \cdot 13 + 1 = 40$$ $$a_4=3 \cdot 40 + 1 = 121$$
Find the next four terms of the sequence given by $a_n=\frac{a_{n-1} - 1}{a_{n-1}}$, $a_0=2$
$$a_0 = 2$$ $$a_1=\frac{2 - 1}{2} = \frac{1}{2}$$ $$a_2=\frac{\frac{1}{2} - 1}{\frac{1}{2}} = -1$$ $$a_3=\frac{-1 - 1}{-1} = 2$$ $$a_4=\frac{2 - 1}{2} = \frac{1}{2}$$
At which point, the sequence becomes cyclic, i.e. it starts to repeat itself.
Find the next four terms of the sequence given by $a_n=a_{n-1} \cdot a_{n-2}$, $a_0=1, a_1=2$
$$a_0 = 1, a_1=2$$ $$a_2=2 \cdot 1 = 2$$ $$a_3=a_2 \cdot 2 = 4$$ $$a_4=4 \cdot 2 = 8$$ $$a_5=8 \cdot 4 = 32$$
Explain why this is equivalent to $F_{k+1}=F_k + F_{k-1}$.
The rule for the Fibonacci sequence is that each term is equal to the sum of the two previous terms. If the "current" term is $k + 1$ then the previous terms are $k$ and $k - 1$ which are the two being added in the revised version of the expression.
Prove that $F_k=3F_{k-3}+2F_{k-4}$, for $k\ge 4$.
Start with $F_k$ and substitute "the sum of the previous two terms" repeatedly until you get the required expression.
Prove that $F_k=3F_{k-3}+2F_{k-4}$, for $k\ge 4$.
First, notice that
$$F_k = F_{k-1}+F_{k-2}$$ $$F_{k-1} = F_{k-2}+F_{k-3}$$ $$F_{k-2} = F_{k-3}+F_{k-4}$$Then we have
$$\begin{align*} F_k &= F_{k-1}+F_{k-2} \\ &= (F_{k-2}+F_{k-3}) + (F_{k-3}+F_{k-4}) \\ &= (F_{k-3}+F_{k-4} + F_{k-3}) + (F_{k-3}+F_{k-4}) \\ &= 3F_{k-3} + 2F_{k-4} \end{align*}$$Prove that $F_k^2-F_{k-1}^2 = F_k F_{k+1}-F_{k-1}F_{k+1}$.
$$\begin{align*} F_k^2-F_{k-1}^2 &= (F_k - F_{k-1})(F_k + F_{k-1}) \\ &= (F_k - F_{k-1})(F_{k+1}) \\ &= F_kF_{k+1} - F_{k-1}F_{k+1} \end{align*} $$
Infer an explicit formula for the sequence $a_n = 6a_{n-1}$, $a_0=1$ by looking at the first several terms.
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| $a_n$ | 1 | 6 | 36 | 216 | 1296 | 7776 | 46656 |
Each value is six times the previous so this appears to be $a_n = 6^n$.
Infer an explicit formula for the sequence $a_n = a_{n-1}+1$, $a_0=0$, by looking at the first several terms.
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| $a_n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
Each value is one more than the previous so this appears to be $a_n = n$.
Infer an explicit formula for the sequence $a_n = a_{n-1}+n$, $a_0=1$, by looking at the first several terms.
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| $a_n$ | 0 | 1 | 3 | 6 | 10 | 15 | 21 |
Each term is equal to the sum of all the previous terms, i.e. $a_n$ is equal to the sum of the first $n$ integers which is given by $\frac{n(n+1)}{2}$.
Suppose a population increases by 3.5% per year. Find a formula that represents the size of the population after $n$ years.
$$\Delta a_n = 0.035a_n$$ $$ a_{n+1} - a_n = 0.035a_n$$ $$ a_{n+1} = 1.035a_n$$
The concentration of a medication in the patient's blood has to stay above 100 mg. An initial dosage has 500 mg. If the concentration in the blood stream reduces by 15% every 6 hours develop a model that describes the concentration after n 6 hour intervals.
$$\Delta a_n = -0.15a_n$$ $$a_{n+1} - a_n = -0.15a_n$$ $$a_{n+1} = 0.85a_n$$ $$a_n = a_0 (0.85)^n = 500 (0.85)^n$$
Give a recursive definition for $\Pi_{i=0}^n a_i$.
$$\Pi_{i=0}^n a_i = a_n \cdot \Pi_{i=0}^{n+1} a_i$$
Show that $\Pi_{i=0}^n (a_i b_i) = \Pi_{i=0}^n a_i \Pi_{i=0}^n b_i$ where $a_i, b_i \in \mathbb{R}$.
Let $P(n): \Pi_{i=0}^n (a_i b_i) = \Pi_{i=0}^n a_i \Pi_{i=0}^n b_i$ be the induction hypothesis.
When $n=0$, $\Pi_{i=0}^0 (a_i b_i) = a_0 b_0$ which is $P(0)$.
Assume $P(n)$ is true. Then
$$\begin{align*} \Pi_{i=0}^{n+1} (a_i b_i) &= a_{n+1} b_{n+1} \Pi_{i=0}^{n} (a_i b_i) \\ &= a_{n+1} b_{n+1} \Pi_{i=0}^{n} a_i \Pi_{i=0}^{n}b_i \\ &= (a_{n+1} \Pi_{i=0}^{n} a_i)( b_{n+1} \Pi_{i=0}^{n}b_i) \\ &= \Pi_{i=0}^{n+1} a_i \Pi_{i=0}^{n+1}b_i \\ \end{align*}$$Which is $P(n+1)$.
An annuity is an investment where you make a single initial payment then the account earns a fixed interest rate every month ($r$) and you get a fixed payment from the account every month ($b$). Develop a difference equation that represents the amount remaining in the account after $n$ months. (This same model can be used to describe the remaining balance on a mortgage payment or credit card.)
$$\Delta a_n = ra_n - b$$ $$a_{n+1} - a_n = ra_n - b$$ $$a_{n+1} = (1+r)a_n - b$$
