Let P(n) be the statement

$$2 ^ { n } > n.$$

Observe that if

$$2 ^ { n } > n,$$

then

$$2 ^ { n } + 2 ^ { n } > 2 n.$$

Use this to show that if P(n) is true for n = k, then P(n) is true for n = k + 1. Conclude that P(n) is true for all n.

Convert each angle in radians to degrees.

$$\frac { 5 \pi } { 12 }$$

A sequence is defined by

$$u _ { n } = \frac { 71 - 7 n } { 2 }$$

. a. Show that the sequence is arithmetic. b. Find

$$u_1$$

and d. c. Find

$$u_{75}$$

. d. For what values of n are the terms of the sequence less than -200?

A particle moves with acceleration a(t) m/

$$s^2$$

along and s-axis and has velocity

$$v_0$$

m/s at time t=0. FInd the displacement and the distance traveled by the particle during the given time interval. a(t)=-2;

$$v_0$$

=3;

$$1 \leq t \leq 4$$

$$\begin{array} { l } { \text { Let } E \text { be an extension of } F \text { and let } a \text { and } b \text { belong to } E . \text { Prove that } } \\ { F ( a , b ) = F ( a ) ( b ) = F ( b ) ( a ) } \end{array}$$

Write the ratio as a fraction in simplest form. 15. 16 dogs to 12 cats

Square ABCD has vertices at (0, 0), (0, 2), (2, 2), and (2, 0). Perform the indicated transformation. Then give the coordinates of figure

$$A ^ { \prime } B ^ { \prime } C ^ { \prime } D ^ { \prime }$$

.

$$( x , y ) \rightarrow ( x + 3 , y )$$

Perform the indicated calculation.

$$_ { 8 } P _ { 6 }$$

(a) Prove that there exist unique scalars $c_0,c_1,\dots,c_n$ such that

$$f(-1)+f(-2)=c_0f(0)+c_1f(1)+\dots+c_nf(n)$$

for every polynomial $f(x)$ in $\mathcal{P}_n$. (b) Find the scalars $c_0,c_1,c_2,c_3,$ and $c_4$ such that

$$f(-1)+f(-2)=c_0f(0)+c_1f(1)+c_2f(2)+c_3f(3)+c_4f(4)$$

for every polynomial $f(x)$ in $\mathcal{P}_4$.

Determine the best method to solve the system of equation. Then solve the system. 6x + 5y = 9, -2x + 4y = 14

Simplify: 8 1/3 · 1 4/5

We now have $5,000 in assets and are given a choice between investment 1 and investment 2. With investment 1, 80% of the time weincrease our asset position by$295,000, and 20% of the time we increase our asset position by $95,000. With investment 2, 50% of the time we increase our asset position by$595,000, and 50% of the time we increase our asset position by $5,000. Our utility function for final asset position x is u(x). We are given the following values for u(x): u(0) = 0, u(640,000) = .80, u(810,000) = .90, u(0) = 0, u(90,000) = .30, u(1,000,000) = 1, u(490,000) = .7. a. Are we risk-averse, risk-seeking, or risk-neutral? Explain. b. Will we prefer investment 1 or investment 2?

Find the: a. smallest multiple of 6 that is greater than 200 b. greatest multiple of 11 that is less than 500.

Juanita runs every third day and swims every fifth day. If Juanita runs and swims today, in how many days will she run and swim again on the same day?