Determine which statements are true and which are false. Prove those that are true and disprove those that are false.

$$\sqrt { 2 } / 6$$

is rational.

The purpose of this exercise is to develop a formula for the monthly payment P that is required to repay a loan of A dollars in n monthly installments, with the interest on the unpaid balance equal to a monthly rate r. Let $A_k=$ amount still owed after paying k installments. Then interest for the (k+1) st month is $r A_k$, and the principal paid off in the (k+1) st payment is $P-r A_k$. Thus,

$$A_{k+1}=A_k-\left(P-r A_k\right)=(1+r) A_k-P .$$

a. Use the equation above and the fact that $A_0=A$ to find $A_1, A_2$, and $A_3$ in terms of A, r, and P.

b. Generalizing from $A_1, A_2$, and $A_3$, you get $A_n=(1+r)^n A-\left[(1+r)^{n-1}+(1+r)^{n-2}+\cdots+(1+r)+1\right] P$.

Since the bracketed quantity above is a geometric series, show that the equation can be rewritten

$$A_n=(1+r)^n A-\left[\frac{(1+r)^n-1}{r}\right] P .$$

c. Since $A_n=0$ (why?), use the last equation in part (b) to show that

$$P=\frac{A r(1+r)^n}{(1+r)^n-1} \text {. }$$

Briefly explain the importance of each of the following. culture

Circle the word or words modified by the adverb in italics. On the blank, write $v$ if the adverb modifies a verb. Write $adj$. if the adverb modifies an adjective. Write $adv$. if the adverb modifies another adverb.\ ______I don’t think she was entirely sure what she had said.

In many applications, it is necessary to write expressions in the form $c x^n$, where $c$ is a constant and $n$ is an integer. Write the expressions in this form.

$$\frac{1}{4 x^2}$$

Infer Two astronomers with telescopes, one on Venus and one on Mars, are observing Earth. Who would have an easier time and why?

Compute the nun1ber of (a) five-digit binary numbers.

Show that the inverse of the "subtracting" function, $f(x)=x-k$, is the "adding" function, $f(x)=x+k$.

Write as an algebraic expression.

Suppose $w$ was Lisa's weight last year. Lisa lost $7$ pounds this year. Write an expression for Lisa's weight this year.

Differentiate the causes of frequent voiding associated with cystitis and with renal insufficiency.

Choose an appropriate data display for the situation. Explain your reasoning. percent of class participating in school clubs

Determine which statements are true and which are false. Prove those that are true and disprove those that are false.

$$3 \sqrt { 2 } - 7$$

is irrational.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. $f(x)=\frac{x-4}{5-x}$

Find the sum of each arithmetic series. The first 100 terms of the series

$$100 + 98 + 96 + 94 + \cdots$$