Chapter 8

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{9} P_{4} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}8 \\ 3\end{array}\right)$$

Write the first five terms of each geometric sequence. $$ a_{1}=5, \quad r=3 $$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$ a_{1}=200, d=20 $$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{7} P_{3} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}7 \\ 2\end{array}\right)$$

Write the first five terms of each geometric sequence. $$ a_{1}=4, \quad r=3 $$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=300, d=50$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{8}P_{5} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}12 \\ 1\end{array}\right)$$

Write the first five terms of each geometric sequence. $$ a_{1}=20, \quad r=\frac{1}{2} $$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=-7, d=4$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{10} P_{4} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}11 \\ 1\end{array}\right)$$

In Exercises \(1-4,\) a statement \(S_{n}\) about the positive integers is given. Write statements \(S_{1}, S_{2},\) and \(S_{3},\) and show that each of these statements is true. \(S_{n}: 3\) is a factor of \(n^{3}-n\).

Write the first five terms of each geometric sequence. $$ a_{1}=24, \quad r=\frac{1}{3} $$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=-8, d=5$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{6} P_{6} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}6 \\ 6\end{array}\right)$$

Write the first five terms of each geometric sequence. $$ a_{n}=-4 a_{n-1}, \quad a_{1}=10 $$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=300, d=-90$$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{l}15 \\\2\end{array}\right)$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{9}P_{9} $$

Write the first five terms of each geometric sequence. $$a_{n}=-3 a_{n-1}, \quad a_{1}=10$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=200, d=-60$$

Write the first four terms of each sequence whose general term is given. $$a_{n}-\left(-\frac{1}{3}\right)^{n}$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{8}P_{0} $$

In Exercises 1-8, evaluate the given binomial coefficient. $$\left(\begin{array}{c}100 \\\2\end{array}\right)$$

Write the first five terms of each geometric sequence. $$a_{n}=-5 a_{n-1}, \quad a_{1}=-6$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{1}=\frac{s}{2}, d=-\frac{1}{2}$$

Use the formula for \(_{n} P_{t}\) to evaluate each expression. $$ _{6} P_{0} $$

Write the first five terms of each geometric sequence. $$a_{n}=-6 a_{n-1}, \quad a_{1}=-2$$

Write the first four terms of each sequence whose general term is given. $$a_{n}-(-1)^{n+1}(n+4)$$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x+2)^{3}$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}+6, a_{1}=-9$$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1, and common ratio, r. Find \(a_{8}\) when \(a_{1}=6, r=2.\)

Write the first four terms of each sequence whose general term is given. $$a_{n}-\frac{2 n}{n+4}$$

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{10} C_{6} $$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(x+4)^{3}$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}+4, a_{1}=-7$$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1, and common ratio, r. Find \(a_{8}\) when \(a_{1}=5, r=3.\)

Write the first four terms of each sequence whose general term is given. $$a_{n}-\frac{3 n}{n+5}$$

A die is rolled. Find the probability of getting a 4

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{11} \mathrm{C}_{4} $$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(3 x+y)^{3}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$4+8+12+\dots+4 n=2 n(n+1)$$

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of each sequence with the given first term, a1, and common ratio, r. Find \(a_{12}\) when \(a_{1}=5, r=-2.\)

Write the first four terms of each sequence whose general term is given. $$a_{n}-\frac{(-1)^{n+1}}{2^{n}-1}$$

A die is rolled. Find the probability of getting a 5

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{12} C_{3} $$