An arithmetic progression has first term a and common difference 10. The sum of the first n terms of the progression is 10 000. Express a in terms of n, and show that the nth term of the progression is $\frac{10000}{n}+5(n-1)$. Given that the nth term is less than 500, show that $n^{2}-101 n+2000<0$ and hence find the largest possible value of n.

A diode has $I_S = 0.2$ fA and n = 1. (a) What is the diode current if the diode voltage is 0.675 V? (b) What will be the diode voltage if the current increases by a factor of 3?

Are all even-degree polynomial functions even? Are all odd-degree polynomial functions odd? Explain.

The sodium nitrate (NaNO$_{3}$) solubility data is described nicely by the regression line y = 67.508 + 0.871x, where s = 0.959. Construct a 90% confidence interval for the y-intercept, $\beta_{0}$.

Recall the cereal company that was putting photos of athletes in boxes of cereal? It puts pictures of Serena Williams in half the boxes and pictures of either LeBron James or Dustin Johnson in the others. Hoping to get a complete set, you buy three boxes of cereal. a) Make a sample space showing the possible outcomes as you open the boxes one at a time. b) Why can't you use this sample space to find the probability that you get one picture of each athlete?

Differentiate the moment-generating function for a geometric random variable and verify the expressions given for E(X) and Var(X) in Theorem 4.4.1.

State what the following abbreviations stand for: NOI, GI, $T_{e},$ NOPAT, TI, R, OE, EBIT.

Find an expression, in terms of n, for the coefficient of x in the expansion $(1+4 x)+(1+4 x)^{2}+(1+4 x)^{3}+\ldots+(1+4 x)^{n}$.

Choose the best answer. Which is NOT true about geothermal energy? (a) It is only available in limited areas. (b) It cannot be used for cooling. (c) It can be locally depleted due to heavy use. (d) Ground source heat pumps require an additional source of energy.

The points A(-3, -4) and C(5, 4) are the ends of the diagonal of a rhombus ABCD. (a) Find the equation of the diagonal BD. (b) Given that the side BC has gradient $\frac{5}{3}$, find the coordinates of B and hence of D.

Suppose that

$$f_{X, Y}(x, y)=\lambda^{2} e^{-\lambda(x+y)}, 0 \leq x, 0 \leq y$$

Find E(X + Y).

Compute $-2 \ln \lambda$ for the nightmare data, and use it to test the hypothesis that $p_{X}=p_{Y} .$ Let $\alpha=0.01$

From 2000 through 2009, the population of Kentucky grew more slowly than that of South Carolina. Models that represent the populations of the two states are given by

$$\left\{\begin{array}{ll} P=30.8 t+4035 & \mathrm { Kentucky } \\ P=61.6 t+3985 & \mathrm { South Carolina } \end{array}\right.$$

where P represents the population (in thousands) and t represents the year, with t=0 corresponding to 2000. Use a graphing utility to estimate the year when the population of South Carolina first exceeded that of Kentucky.

A restaurant offers a choice of four appetizers, four-teen entrees, six desserts, and five beverages. How many different meals are possible if a diner intends to order only three courses? (Consider the beverage to be a “course.”)