Chapter 9

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 \(n.\) $$a_{1}=30, a_{k+1}=-\frac{2}{3} a_{k}$$

Use the Binomial Theorem to expand and simplify the expression. \((2 x-y)^{5}\)

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

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=5, d=6$$

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.) Neither marble is yellow.

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=11, r=1.03,12 \mathrm{th}\) term

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

Use the table feature of a graphing utility to find the first 10 terms of the sequences. (Assume \(n\) begins with 1.) $$a_{n}=(-1)^{n+1}+8$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=5, d=-\frac{3}{4}$$

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{5} P_{5}$$

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.) The marbles are of different colors.

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=24, r=2.6,8\)th term

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

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{n^{2}}{n^{2}+1}\\\ &a_{10}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=-2.6, d=0.2$$

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{8} P_{3}$$

The complement of an event \(A\) is the collection of all outcomes in the sample space that are not in \(A\). If the probability of \(A\) is \(P(A),\) then the probability of the complement \(A^{\prime}\) is given by \(P\left(A^{\prime}\right)=1-P(A) .\) You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=0.75$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=8, r=-\frac{4}{3}, 7 \mathrm{th}\) term

Use the Binomial Theorem to expand and simplify the expression. \((2 y-5)^{3}\)

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{n^{2}}{2 n+1}\\\ &a_{5}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{1}=-10, d=9$$

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{20} P_{2}$$

The complement of an event \(A\) is the collection of all outcomes in the sample space that are not in \(A\). If the probability of \(A\) is \(P(A),\) then the probability of the complement \(A^{\prime}\) is given by \(P\left(A^{\prime}\right)=1-P(A) .\) You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=0.36$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=8, r=-\frac{3}{4}, 9\)th term

Use the Binomial Theorem to expand and simplify the expression. \((2 r-3 s)^{6}\)

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=(-1)^{n}(3 n-2)\\\ &a_{25}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{8}=26, a_{12}=42$$

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{6} P_{5}$$

The complement of an event \(A\) is the collection of all outcomes in the sample space that are not in \(A\). If the probability of \(A\) is \(P(A),\) then the probability of the complement \(A^{\prime}\) is given by \(P\left(A^{\prime}\right)=1-P(A) .\) You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=\frac{2}{3}$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=-\frac{1}{4}, r=8,6\)th term

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

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=(-1)^{n-1}[n(n-1)]\\\ &a_{16}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{6}=-38, a_{11}=-73$$

Evaluate \(_{n} P_{r}\) using the formula from this section. $$_{7} P_{4}$$

The complement of an event \(A\) is the collection of all outcomes in the sample space that are not in \(A\). If the probability of \(A\) is \(P(A),\) then the probability of the complement \(A^{\prime}\) is given by \(P\left(A^{\prime}\right)=1-P(A) .\) You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=\frac{7}{8}$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=-\frac{1}{128}, r=2,12 \mathrm{th}\) term

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

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{2^{n+1}}{2^{n}+1}\\\ &a_{7}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{3}=19, a_{15}=-1.7$$

Evaluate \(_{n} P_{r}\) using a graphing utility. $$_{30} P_{6}$$

You are given the probability that an event will not happen. Find the probability that the event will happen. $$P\left(E^{\prime}\right)=0.12$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=7, r=\sqrt{2}, 14 \mathrm{th}\) term

Use the Binomial Theorem to expand and simplify the expression. \(\left(y^{2}+2\right)^{6}\)

Find the indicated term of the sequence. $$\begin{aligned} &a_{n}=\frac{3^{n}}{3^{n}+1}\\\ &a_{6}= \end{aligned}$$

Write the first five terms of the arithmetic sequence. Use the table feature of a graphing utility to verify your results. $$a_{5}=16, a_{14}=38.5$$

You are given the probability that an event will not happen. Find the probability that the event will happen. $$P\left(E^{\prime}\right)=0.84$$

Finding a Term of a Geometric Sequence Find the indicated term of the geometric sequence (a) using the table feature of a graphing utility and (b) algebraically. \(a_{1}=2, r=\sqrt{3}, 11 \mathrm{th}\) term

Write an expression for the apparent \(n\) th term of the sequence. (Assume \(n\) begins with \(1 .\)) $$3,8,13,18,23, \dots$$

Write the first five terms of the arithmetic sequence. Find the common difference and write the \(n\) th term of the sequence as a function of \(n .\) $$a_{1}=15, a_{k+1}=a_{k}+4$$

Evaluate \(_{n} P_{r}\) using a graphing utility. $$_{120} P_{4}$$