Chapter 8

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

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$3+4+5+\cdots+(n+2)=\frac{n(n+5)}{2}$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}-20, a_{1}=50$$

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 \(a_{1}=4, r=-2.\)

A die is rolled. Find the probability of getting an odd number.

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{7} C_{7} $$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(5 x-1)^{3}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1+3+5+\dots+(2 n-1)=n^{2}$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}-0.4, a_{1}=1.6$$

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 \(a_{40}\) when \(a_{1}=1000, r=-\frac{1}{2}.\)

A die is rolled. Find the probability of getting a number greater than 3

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

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(4 x-1)^{3}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$3+6+9+\dots+3 n=\frac{3 n(n+1)}{2}$$

In Exercises \(1-14\), write the first six terms of cach arithmetic sequence $$a_{n}=a_{n-1}-03, a_{1}=-1.7$$

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 \(a_{30}\) when \(a_{1}=8000, r=-\frac{1}{2}.\)

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

Use the formula for \(_{n} C_{r}\) to evaluate each expression. $$ _{5} C_{0} $$

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(2 x+1)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$3+7+11+\dots+(4 n-1)=n(2 n+1)$$

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{6}\) when \(a_{1}=13, d=4\)

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 \(a_{1}=1,000,000, r=0.1.\)

A die is rolled. Find the probability of getting a number greater than 7.

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

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$(3 x+1)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$2+7+12+\dots+(5 n-3)=\frac{n(5 n-1)}{2}$$

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{16}\) when \(a_{1}=9, d=2\)

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 \(a_{\mathrm{s}}\) when \(a_{1}=40,000, r=0.1.\)

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a queen.

Does the problem involve permutations or combinations? Explain your answer (It is not necessary to solve the problem. A medical researcher needs 6 people to test the effectiveness of an experimental drug. If 13 people have volunteered for The test, in how many ways can 6 people be selected?

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(x^{2}+2 y\right)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1+2+2^{2}+\cdots+2^{n-1}=2^{n}-1$$

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{50}\) when \(a_{1}=7, d=5\)

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$ 3,12,48,192, \dots $$

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a diamond.

Does the problem involve permutations or combinations? Explain your answer (It is not necessary to solve the problem. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If first prize is \(\$ 1000,\) second prize is \(\$ 500\) and third prize is \(\$ 100,\) in how many different ways can the prizes be awarded?

In Exercises 9-30, use the Binomial Theorem to expand each binomial and express the result in simplified form. $$\left(x^{2}+y\right)^{4}$$

In Exercises 11-24, use mathematical induction to prove that each statement is true for every positive integer \(n.\) $$1+3+3^{2}+\dots+3^{n-1}=\frac{3^{n}-1}{2}$$

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{60}\) when \(a_{1}=8, d=6\)

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$3,15,75,375, \dots$$

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a picture card.

Does the problem involve permutations or combinations? Explain your answer (It is not necessary to solve the problem. How many different four-letter passwords can be formed from the letters \(A, B, C, D, E, F,\) and \(G\) if no repetition of letters is allowed?

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

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

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{200}\) when \(a_{1}=-40, d=5\)

Write a formula for the general term (the nth term of each geometric sequence. Then use the formula for \(a_{n}\) to find \(a_{7},\) the seventh term of the sequence. $$18,6,2, \frac{2}{3}, \dots$$

The general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. $$a_{n}-\frac{n^{2}}{n !}$$

You are dealt one card from a standard 52-card deck. Find the probability of being dealt a card greater than 3 and less than 7 .

In Exercises \(15-22,\) find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\) Find \(a_{150}\) when \(a_{1}=-60, d=5\)

Does the problem involve permutations or combinations? Explain your answer (It is not necessary to solve the problem. Fifty people purchase raffle tickets. Three winning tickets are selected at random. If each prize is \(\$ 500,\) in how many different ways can the prizes be awarded?