91Ó°ÊÓ

Comparing Graphs of a Sequence and a Line (a) Graph the first 10 terms of the arithmetic sequence \(a_{n}=2+3 n.\) (b) Graph the equation of the line \(y=3 x+2.\) (c) Discuss any differences between the graph of \(a_{n}=2+3 n\) and the graph of \(y=3 x+2.\) (d) Compare the slope of the line in part (b) with the common difference of the sequence in part (a). What can you conclude about the slope of a line and the common difference of an arithmetic sequence?

Fnd the sum of the infinite series. $$\sum^{\infty} \frac{6}{10 i}$$

An investment firm has a job opening with a salary of \(\$ 45,000\) for the first year. During the next 39 years, there is a \(5 \%\) raise each year. Find the total compensation over the 40 -year period.

In your own words, explain how to form the rows of Pascal's Triangle.

Prove the property for all integers \(r\) and \(n\) where \(0 \leq r \leq n\).The sum of the numbers in the \(n\) th row of Pascal's Triangle is \(2^{n}\).

\(A 3 \times 3 \times 3\) cube is made up of 27 unit cubes (a unit cube has a length, width, and height of 1 unit), and only the faces of each cube that are visible are painted blue, as shown in the figure. (a) Complete the table to determine how many unit cubes of the \(3 \times 3 \times 3\) cube have 0 blue faces, 1 blue face, 2 blue faces, and 3 blue faces. $$\begin{array}{|l|l|l|l|l|} \hline \begin{array}{l} \text { Number of } \\ \text { Blue Cube Faces } \end{array} & 0 & 1 & 2 & 3 \\ \hline 3 \times 3 \times 3 & & & & \\ \hline \end{array}$$ (b) Repeat part (a) for a \(4 \times 4 \times 4\) cube, a \(5 \times 5 \times 5\) cube, and a \(6 \times 6 \times 6\) cube. (c) What type of pattern do you observe? (d) Write formulas you could use to repeat part (a) for an \(n \times n \times n\) cube.