5.1 Sequences  (Page 10/25)

$n\text{/}{2}^n,$ $n\ge 2$

$\text{ln}\left(1+\frac{1}{n}\right)$

$\text{sin}n$

$\text{cos}\left(n^2\right)$

$n^{1\text{/}n},$ $n\ge 3$

$n^{\mathrm{-1}\text{/}n},$ $n\ge 3$

$\text{tan}n$

Determine whether the sequence defined as follows has a limit. If it does, find the limit.

$a_1=\sqrt{2},$ $a_2=\sqrt{2\sqrt{2}},$ $a_3=\sqrt{2\sqrt{2\sqrt{2}}}$ etc.

Determine whether the sequence defined as follows has a limit. If it does, find the limit.

$a_1=3,$ $a_n=\sqrt{2a_{n-1}},$ $n=2,3\text{,…}.$

$n\text{sin}\left(1\text{/}n\right)$

$\frac{\text{cos}\left(1\text{/}n\right)-1}{1\text{/}n}$

$a_n=\frac{n\text{!}}{n^n}$

$a_n=\text{sin}n\text{sin}\left(1\text{/}n\right)$

[T] $a_n=\text{sin}n$

[T] $a_n=\text{cos}n$

$a_n={\text{tan}}^{\mathrm{-1}}(n^2)$

$a_n={(2n)}^{1\text{/}n}-n^{1\text{/}n}$

$a_n=\frac{\text{ln}(n^2)}{\text{ln}(2n)}$

$a_n={\left(1-\frac{2}{n}\right)}^n$

$a_n=\text{ln}\left(\frac{n+2}{n^2-3}\right)$

$a_n=\frac{2^n+3^n}{4^n}$

$a_n=\frac{{(1000)}^n}{n\text{!}}$

$a_n=\frac{{(n\text{!})}^2}{(2n)\text{!}}$

[T] $f\left(x\right)=x^2-2,$ $x_0=1$

[T] $f\left(x\right)={\left(x-1\right)}^2-2,$ $x_0=2$

[T] $f\left(x\right)=e^x-2,$ $x_0=1$

[T] $f\left(x\right)=\text{ln}x-1,$ $x_0=2$

[T] Suppose you start with one liter of vinegar and repeatedly remove $0.1\text{L,}$ replace with water, mix, and repeat.

1. Find a formula for the concentration after $n$ steps.
2. After how many steps does the mixture contain less than $10\text{\%}$ vinegar?

[T] A lake initially contains $2000$ fish. Suppose that in the absence of predators or other causes of removal, the fish population increases by $6\text{\%}$ each month. However, factoring in all causes, $150$ fish are lost each month.

1. Explain why the fish population after $n$ months is modeled by $P_n=1.06P_{n-1}-150$ with $P_0=2000.$
2. How many fish will be in the pond after one year?

[T] A bank account earns $5\text{\%}$ interest compounded monthly. Suppose that $\text{\$}1000$ is initially deposited into the account, but that $\text{\$}10$ is withdrawn each month.

1. Show that the amount in the account after $n$ months is $A_n=\left(1+.05\text{/}12\right)A_{n-1}-10;$ $A_0=1000.$
2. How much money will be in the account after $1$ year?
3. Is the amount increasing or decreasing?
4. Suppose that instead of $\text{\$}10,$ a fixed amount $d$ dollars is withdrawn each month. Find a value of $d$ such that the amount in the account after each month remains $\text{\$}1000.$
5. What happens if $d$ is greater than this amount?

[T] A student takes out a college loan of $\text{\$}\mathrm{10,000}$ at an annual percentage rate of $6\text{\%},$ compounded monthly.

1. If the student makes payments of $\text{\$}100$ per month, how much does the student owe after $12$ months?
2. After how many months will the loan be paid off?

[T] Consider a series combining geometric growth and arithmetic decrease. Let $a_1=1.$ Fix $a>1$ and $0<b<a.$ Set $a_{n+1}=a.a_n-b.$ Find a formula for $a_{n+1}$ in terms of $a^n,$ $a,$ and $b$ and a relationship between $a$ and $b$ such that $a_n$ converges.

[T] The binary representation $x=0.b_1{b}_2{b}_3...$ of a number $x$ between $0$ and $1$ can be defined as follows. Let $b_1=0$ if $x<1\text{/}2$ and $b_1=1$ if $1\text{/}2\le x<1.$ Let $x_1=2x-b_1.$ Let $b_2=0$ if $x_1<1\text{/}2$ and $b_2=1$ if $1\text{/}2\le x<1.$ Let $x_2=2x_1-b_2$ and in general, $x_n=2x_{n-1}-b_n$ and $b_{n-1}=0$ if $x_n<1\text{/}2$ and $b_{n-1}=1$ if $1\text{/}2\le x_n<1.$ Find the binary expansion of $1\text{/}3.$

[T] To find an approximation for $\pi ,$ set $a_0=\sqrt{2+1},$ $a_1=\sqrt{2+a_0},$ and, in general, $a_{n+1}=\sqrt{2+a_n}.$ Finally, set $p_n={3.2}^n\sqrt{2-a_n}.$ Find the first ten terms of $p_n$ and compare the values to $\pi .$

[T] Starting with $K=\mathrm{16,807}$ and $M=\mathrm{2,147,483,647},$ using ten different starting values of $a_1,$ compute sequences of bits $b_n$ up to $n=1000,$ and compare their averages to ten such sequences generated by a random bit generator.

[T] Find the first $1000$ digits of $\pi$ using either a computer program or Internet resource. Create a bit sequence $b_n$ by letting $b_n=1$ if the $n\text{th}$ digit of $\pi$ is odd and $b_n=0$ if the $n\text{th}$ digit of $\pi$ is even. Compute the average value of $b_n$ and the average value of $d_n=\left|b_{n+1}-b_n\right|,$ $n=1\text{,...},999.$ Does the sequence $b_n$ appear random? Do the differences between successive elements of $b_n$ appear random?