(a) Prove the result in the earlier example by induction.

(b) Prove the result of the earlier example directly by recursively substituting $(1+x^k )= (1−x^{2k} )/(1−x^k)$ in g*(x).

Write as an ordinary decimal number: a. The estimated population of the world in the year 2010 is

$$6.8 \times 10 ^ { 9 }$$

people. b. 80 gsm photocopy paper has a thickness of

$$1.1 \times 10 ^ { - 4 }$$

metres. c. A small virus has greatest width of

$$9.8 \times 10 ^ { - 4 } \mathrm { mm }$$

. d. The diameter of the Milky Way is

$$1.4 \times 10 ^ { 5 }$$

light years.

What is the basic particle of an element called?

The subjunctive mood of a verb states a wish or a condition contrary to fact. The subjunctive is also used in $that$ clauses of recommendation, resolution, command, or demand. The subjunctive form appears in - Certain forms of the verb to $be$ - the present is $be$ and the past is $were$- The third-person singular of other verbs, usually used without the final $s$\Condition Contrary to Fact: If you were me and I were you ... I wonder what we should do! (form of to be)\That Clause of Command: Paul's mother insisted that he stop rocking. (third-person singular without $-s$ )\Practice Identify the correct verb for given sentence.
"It had seemed for a moment as if the housemaster (was, were) going to laugh".

Choose the correct description of the skin condition.

_______ urticaria

a.an autoimmune skin condition

b.skin cancer that can develop from a mole; the most serious form of skin cancer

c.another name for hives

d.skin lesion caused by a shallow scratch

e.another name for a skin boil

f.a virus-caused skin cancer that sometimes develops in immune-deficient individuals

Why is the genome of the tube worm symbiont thought to be so much larger than the genomes of insect symbionts?

Kim-Ly Vanh's bank granted her a single-payment loan of $2,580 for 90 days at 12% ordinary interest. What is the amount of interest? A.$127.44 B. $77.40 C.$87.54 D. $98.12

Find the number of subsets and the number of proper subsets for each set. {x|x is a month that begins with the letter J}

Rodrigo practiced playing the guitar

$$15\frac{1}{3}$$

hours over the past 3 weeks. He practiced for

$$6\frac{1}{4}$$

hours during the first week and

$$4\frac{2}{3}$$

hours during the second week. How much time did Rodrigo spend practicing during the third week? Use the numbers and symbols to write an equation that represents the problem. Then solve the equation. Symbols may be used more than once or not at all.

$$15 \frac { 1 } { 3 }$$

$$6 \frac { 1 } { 4 }$$

$$4 \frac { 2 } { 3 }$$

$$\times$$

$$=$$

$$+$$

Practice time during third week: ______ hours.

n each of the following sentences, circle any letter that should be capitalized and draw a slash (/) through any letter that is capitalized but should not be. Example: $\text{\textcircled}{\text{n}}$igetl enjoys stories about $\cancel{\text{T}}$he $\cancel{\text{L}}$egendary $\text{\textcircled}{\text{k}}$ing Arthur of $\text{\textcircled}{\text{b}}$ritain and his knights the letter I found in my Great-Grandmother's papers began, "my dearest mimi."

For $2004$ through $2008$, Coca-Cola annual operating revenues and net incomes were as shown below.

$$\begin{array}{ccc} \text { Year } & \begin{array}{c} x=\text { Operating } \\ \text { Revenue } \end{array} & \begin{array}{c} \boldsymbol{y}=\text { Net } \\ \text { Income } \end{array} \\ \hline 2004 & \$ 21.74 \text { billion } & \$ 4.85 \text { billion } \\ 2005 & 23.10 & 4.87 \\ 2006 & 24.09 & 5.08 \\ 2007 & 28.86 & 5.98 \\ 2008 & 31.94 & 5.81 \end{array}$$

Determine the least-squares regression line and interpret its slope. Estimate net income if the operating revenue is $30 billion.

Consider pressurized liquid water flowing at $\dot{m}=0.1 \mathrm{kg} / \mathrm{s}$ in a circular tube of diameter D=0.1 m and length L=6m. (a) If the water enters at $T_{m, i}=500 \mathrm{K}$ and the surface temperature of the tube is $T_{s}=510 \mathrm{K}$, determine the water outlet temperature $T_{m, o}$. (b) If the water enters at $T_{m, i}=300 \mathrm{K}$ and the surface temperature of the tube is $T_{s}=310 \mathrm{K}$, determine the water outlet temperature $T_{m, o}$. (c) If the water enters at $T_{m, i}=300 \mathrm{K}$ and the surface temperature of the tube is $T_{s}=647 \mathrm{K}$, discuss whether the flow is laminar or turbulent.

Determine whether the statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.

The polynomial $16 x^2+20 x+25$ is a perfect square trinomial.

The practice of harvesting fish faster than they can reproduce is called__________________________________