Chapter 9

Use a graphing utility to find \(_{n} C_{r^{*}}\) \(_{500}{C}_{498}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=\frac{3^{n}}{4^{n}}$$

Write the first five terms of the sequence. Determine whether or not the sequence is arithmetic. If it is, find the common difference. (Assume \(n\) begins with 1.) $$a_{n}=(-1)^{2 n+1}$$

Find the probability for the experiment of selecting one card at random from a standard deck of 52 playing cards. The card is a numbered card \((2-10)\)

In Pennsylvania, each standard automobile license plate number consists of three letters followed by a four-digit number. How many distinct Pennsylvania license plate numbers can be formed?

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=5, r=-\frac{1}{10}$$

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=\frac{1}{n^{3 / 2}}$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=1, d=6$$

In \(1963,\) the United States Postal Service launched the Zoning Improvement Plan (ZIP) Code to streamline the mail-delivery system. A ZIP code consists of a five-digit sequence of numbers. (a) Find the number of ZIP codes consisting of five digits. (b) Find the number of ZIP codes consisting of five digits when the first digit is 1 or 2.

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=6, r=-\frac{1}{4}$$

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=\frac{1}{\sqrt{n}}$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=15, d=4$$

In \(1983,\) in order to identify small geographic segments within a delivery code, the post office began to use an expanded ZIP code called \(\mathrm{ZIP}+4,\) which is composed of the original five-digit code plus a four-digit add-on code. (a) Find the number of ZIP codes consisting of five digits followed by the four additional digits. (b) Find the number of ZIP codes consisting of five digits followed by the four additional digits when the first number of the five-digit code is \(1\) or \(2 .\)

Use the Binomial Theorem to expand and simplify the expression. \((x+1)^{5}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=\frac{(-1)^{n}}{n^{2}}$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=43, d=-7$$

ATM personal identification number (PIN) codes typically consist of four-digit sequences of numbers. (a) Find the total number of ATM codes possible. (b) Find the total number of ATM codes possible when the first digit is not \(0 .\)

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=9, r=e$$

Use the Binomial Theorem to expand and simplify the expression. \((x+1)^{6}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=(-1)^{n}\left(\frac{n}{n+1}\right)$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=100, d=-8$$

Typically, radio stations are identified by four "call letters." Radio stations east of the Mississippi River have call letters that start with the letter \(\mathrm{W},\) and radio stations west of the Mississippi River have call letters that start with the letter \(\mathrm{K}.\) (a) Find the number of different sets of radio station call letters that are possible in the United States. (b) Find the number of different sets of radio station call letters that are possible when the call letters must include a \(\mathrm{Q}.\)

Writing the Terms of a Geometric Sequence Write the first five terms of the geometric sequence. $$a_{1}=7, r=\sqrt{5}$$

Use the Binomial Theorem to expand and simplify the expression. \((a+3)^{3}\)

Write the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. (Assume \(n\) begins with 1.) $$a_{n}=(2 n-1)(2 n+1)$$

Three couples have reserved seats in a row for a concert. In how many different ways can they be seated when (a) there are no seating restrictions? (b) each couple sits together?

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=64, a_{k+1}=\frac{1}{2} a_{k}$$

Use the Binomial Theorem to expand and simplify the expression. \((a+2)^{4}\)

Find a formula for \(a_{n}\) for the arithmetic sequence. $$4, \frac{3}{2},-1,-\frac{7}{2}, . .$$

In how many orders can five girls and three boys walk through a doorway single file when (a) there are no restrictions? (b) the girls walk through before the boys?

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=81, a_{k+1}=\frac{1}{3} a_{k}$$

Use the Binomial Theorem to expand and simplify the expression. \((y-4)^{3}\)

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=2(3 n-1)+5$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=-5, a_{4}=22$$

In how many ways can five children posing for a photograph line up in a row?

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=9, a_{k+1}=2 a_{k}$$

Use the Binomial Theorem to expand and simplify the expression.\((y-5)^{4}\)

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=2 n(n+1)-7$$

Find a formula for \(a_{n}\) for the arithmetic sequence. $$a_{1}=-4, a_{5}=16$$

Design In how many ways can eight people sit in an eight-passenger vehicle?

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=5, a_{k+1}=3 a_{k}$$

Use the Binomial Theorem to expand and simplify the expression. \((x+y)^{5}\)

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=1+\frac{n+1}{n}$$

The nine justices of the U.S. Supreme Court pose for a photograph while standing in a straight line, as opposed to the typical pose of two rows. How many different orders of the justices are possible for this photograph?

Find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. (Hint: Use combinations to find the numbers of outcomes for the given event and sample space.) Both marbles are red.

Finding the \(n\)th Term of a Geometric Sequence Write the first five terms of the geometric sequence. Find the common ratio and write the \(n\)th term of the sequence as a function of \(n.\) $$a_{1}=6, a_{k+1}=-\frac{3}{2} a_{k}$$

Use the Binomial Theorem to expand and simplify the expression. \((x+y)^{6}\)

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=\frac{4 n^{2}}{n+2}$$

Four processes are involved in assembling a product, and they can be performed in any order. The management wants to test each order to determine which is the least time-consuming. How many different orders will have to be tested?

Find the probability for the experiment of drawing two marbles at random (without replacement) from a bag containing one green, two yellow, and three red marbles. (Hint: Use combinations to find the numbers of outcomes for the given event and sample space.) Both marbles are yellow.