Chapter 8

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l}8 \\\3\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{9} P_{4} $$

In Exercises \(1-4,\) a statement \(S_{n}\) about the positive integers is given. Write statements \(S_{1}, S_{2},\) and \(S_{3}\). $$ S_{n}: 1+3+5+\cdots+(2 n-1)=n^{2} $$

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

Write the first six terms of each arithmetic sequence. $$a_{1}=200, d=20$$

Write the first four terms of each sequence whose general term is given. $$a_{n}=3 n+2$$

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l}7 \\\2\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{7} P_{3} $$

In Exercises \(1-4,\) a statement \(S_{n}\) about the positive integers is given. Write statements \(S_{1}, S_{2},\) and \(S_{3}\). $$ S_{n}: 3+4+5+\cdots+(n+2)=\frac{n(n+5)}{2} $$

Write the first six terms of each arithmetic sequence. $$a_{1}=300, d=50$$

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}12 \\\1\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{8} P_{5} $$

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

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

Write the first six terms of each arithmetic sequence. $$a_{1}=-7, d=4$$

Write the first four terms of each sequence whose general term is given. $$a_{n}=3^{n}$$

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}11 \\\1\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{10} P_{4} $$

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

Write the first six terms of each arithmetic sequence. $$a_{1}=-8, d=5$$

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{l}6 \\\6\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{6} P_{6} $$

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

Write the first six terms of each arithmetic sequence. $$a_{1}=300, d=-90$$

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}15 \\\2\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{9} P_{9} $$

Write the first six terms of each arithmetic sequence. $$a_{1}=200, d=-60$$

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}100 \\\2\end{array}\right) $$

In Exercises \(5-10,\) a statement \(S_{n}\) about the positive integers is given. Write statements \(S_{k}\) and \(S_{k+1}\), simplifying statement \(S_{k+1}\) completely. $$ S_{n}: 3+7+11+\cdots+(4 n-1)=n(2 n+1) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{8} P_{0} $$

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

Write the first six terms of each arithmetic sequence. $$a_{1}=\frac{5}{2}, d=-\frac{1}{2}$$

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

Evaluate the given binomial coefficient. $$ \left(\begin{array}{c}100 \\\98\end{array}\right) $$

In Exercises \(1-8,\) use the formula for \(_{n} P_{r}\) to evaluate each expression. $$ _{6} P_{0} $$

Write the first six terms of each arithmetic sequence. $$a_{1}=\frac{3}{4}, d=-\frac{1}{4}$$

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

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+2)^{3} $$

The sample space of equally likely outcomes is \(\\{1,2,3,4,5,6\\} .\) Find the probability of getting: a 4.

In Exercises \(9-16,\) use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{9} C_{5} $$

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

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (x+4)^{3} $$

The sample space of equally likely outcomes is \(\\{1,2,3,4,5,6\\} .\) Find the probability of getting: a 5.

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, \(a_{1},\) 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. $$a_{n}=\frac{3 n}{n+5}$$

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x+y)^{3} $$