Chapter 9

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{2^{n}}{n^{3}} $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}25 \\ 11\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many outcomes are possible from tossing a pair of coins?

For the following exercises, express each arithmetic sum using summation notation. \(5+10+15+20+25+30+35+40+45+50\)

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(5,5.2,5.4,5.6,5.8, \ldots\)

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. \(a_{1}=-25, d=-9\)

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{2 n+1}{n^{3}} $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{c}17 \\ 6\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many outcomes are possible from tossing a coin and rolling a 6-sided die?

For the following exercises, express each arithmetic sum using summation notation. \(10+18+26+\ldots+162\)

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(-1, \frac{1}{2},-\frac{1}{4}, \frac{1}{8},-\frac{1}{16}, \ldots\)

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. \(a_{1}=0, d=\frac{2}{3}\)

For the following exercises, write the first four terms of the sequence. $$ a_{n}=1.25 \cdot(-4)^{n-1} $$

For the following exercises, evaluate the binomial coefficient. \(\left(\begin{array}{l}200 \\ 199\end{array}\right)\)

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many two-letter strings \(-\) the first letter from \(A\) and the second letter from \(B-\) can be formed from the sets \(A=\\{b, c, d\\}\) and \(B=\\{a, e, i, o, u\\} ?\)

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(6,8,11,15,20, \ldots\)

For the following exercises, write the first five terms of the arithmetic series given two terms. \(a_{1}=17, \quad a_{7}=-31\)

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-4 \cdot(-6)^{n-1} $$

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 a-b)^{3} $$

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many ways are there to construct a string of 3 digits if numbers can be repeated?

For the following exercises, determine whether the sequence is geometric. If so, find the common ratio. \(0.8,4,20,100,500, \ldots\)

For the following exercises, write the first five terms of the arithmetic series given two terms. \(a_{13}=-60, \quad a_{33}=-160\)

For the following exercises, write the first four terms of the sequence. $$ a_{n}=\frac{n^{2}}{2 n+1} $$

For the following exercises, two coins are tossed. What is the sample space?

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (5 a+2)^{3} $$

For the following exercises, determine whether to use the Addition Principle or the Multiplication Principle. Then perform the calculations. How many ways are there to construct a string of 3 digits if numbers cannot be repeated?

For the following exercises, use the formula for the sum of the first \(n\) terms of each arithmetic sequence. \(19+25+31+\ldots+73\)

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. \(a_{1}=8, \quad r=0.3\)

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is \(3,\) common difference is 4, find the \(5^{\text {th }}\) term.

For the following exercises, write the first four terms of the sequence. $$ a_{n}=(-10)^{n}+1 $$

For the following exercises, two coins are tossed. Find the probability of tossing two heads.

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (3 a+2 b)^{3} $$

For the following exercises, compute the value of the expression. $$ P(5,2) $$

For the following exercises, write the first five terms of the geometric sequence, given the first term and common ratio. \(a_{1}=5, \quad r=\frac{1}{5}\)

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is 4, common difference is \(5,\) find the \(4^{\text {th }}\) term.

For the following exercises, write the first four terms of the sequence. $$ a_{n}=-\left(\frac{4 \cdot(-5)^{n-1}}{5}\right) $$

For the following exercises, two coins are tossed. Find the probability of tossing exactly one tail.

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (2 x+3 y)^{4} $$

For the following exercises, compute the value of the expression. $$ P(8,4) $$

For the following exercises, express each geometric sum using summation notation. \(1+3+9+27+81+243+729+2187\)

For the following exercises, write the first five terms of the geometric sequence, given any two terms. \(a_{7}=64, \quad a_{10}=512\)

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is 5, common difference is \(6,\) find the \(8^{\text {th }}\) term.

For the following exercises, write the first eight terms of the piecewise sequence. \(a_{n}=\left\\{\begin{array}{ll}(-2)^{n}-2 & \text { if } n \text { is even } \\\ (3)^{n-1} & \text { if } n \text { is odd }\end{array}\right.\)

For the following exercises, two coins are tossed. Find the probability of tossing at least one tail.

For the following exercises, use the Binomial Theorem to expand each binomial. $$ (4 x+2 y)^{5} $$

For the following exercises, compute the value of the expression. $$ P(3,3) $$

For the following exercises, express each geometric sum using summation notation. \(8+4+2+\ldots+0.125\)

For the following exercises, write the first five terms of the geometric sequence, given any two terms. \(a_{6}=25, \quad a_{8}=6.25\)

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. First term is \(6,\) common difference is \(7,\) find the \(6^{\text {th }}\) term.

For the following exercises, write the first eight terms of the piecewise sequence. \(a_{n}=\left\\{\begin{array}{ll}\frac{n^{2}}{2 n+1} & \text { if } n \quad \leq 5 \\ n^{2}-5 & \text { if } n>5\end{array}\right.\)