Chapter 9

Use a graphing utility to graph the first 10 terms of the sequence. (Assume \(n\) begins with 1.) $$a_{n}=\frac{3 n^{2}}{n^{2}+1}$$

Can your calculator evaluate \(_{100} P_{80} ?\) If not, explain why.

Expand the expression in the difference quotient and simplify. \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) \(f(x)=x^{6}\)

Find the sum. $$\sum_{i=1}^{5}(2 i+1)$$

Decide whether each scenario should be counted using permutations or combinations. Explain your reasoning. (Do not calculate.) (a) Number of ways 10 people can line up in a row for concert tickets (b) Number of different arrangements of three types of flowers from an array of 20 types

Writing a Repeating Decimal as a Rational Number Find the rational number representation of the repeating decimal. $$1.38$$

Expand the expression in the difference quotient and simplify. \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) \(f(x)=x^{8}\)

Find the sum. $$\sum_{i=1}^{6}(3 i-1)$$

Determine whether the statement is true or false. Justify your answer. Given the \(n\) th term and the common difference of an arithmetic sequence, it is possible to find the \((n+1)\) th term.

Explain in your own words the meaning of \(_{n} P_{r}\).

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$8+16+32+64+\cdots$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((1+i)^{4}\)

Find the sum. $$\sum_{i=0}^{6} 4 i^{2}$$

Without calculating, determine whether the value of \(_{n} P_{r}\) is greater than the value of \(_{n} C_{r}\) for the values of \(n\) and \(r\) given in the table. Complete the table using yes (Y) or no (N). Is the value of \(_{n} P_{r}\) always greater than the value of \(_{n} C_{r} ?\) Explain.

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$17+14+11+8+\cdot \cdot \cdot \cdot$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((1-i)^{6}\)

Find the sum. $$\sum_{i=0}^{5} 3 i^{2}$$

Prove the identity. $$_{n} P_{n-1}=_{n}P_{n}$$

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$90+30+10+\frac{10}{3}+\cdots$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((4+i)^{4}\)

Find the sum. $$\sum_{j=3}^{5} \frac{1}{j^{2}-3}$$

Prove the identity. $$_{n} C_{n}=_{n} C_{0}$$

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$\frac{5}{4}+\frac{7}{4}+\frac{9}{4}+\frac{11}{4}+\cdots$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((2+i)^{5}\)

Find the sum. $$\sum_{j=3}^{5} \frac{1}{j+1}$$

Explain how to use the first two terms of an arithmetic sequence to find the \(n\) th term.

Prove the identity. $$_{n} C_{n-1}=_{n} C_{1}$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) $$(2-3 i)^{6}$$

Find the sum. $$\sum_{k=1}^{4} 10$$

It The sum of the first 20 terms of an arithmetic sequence with a common difference of 3 is \(650 .\) Find the first term.

Prove the identity. $$_{n} C_{r}=\frac{_{n} P_{r}}{r !}$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((3-2 i)^{6}\)

Find the sum. $$\sum_{k=1}^{5} 4$$

About It The sum of the first \(n\) terms of an arithmetic sequence with first term \(a_{1}\) and common difference \(d\) is \(S_{n} .\) Determine the sum when each term is increased by \(5 .\) Explain.

Use Cramer's Rule to solve the system of equations. $$\left\\{\begin{array}{r} -5 x+3 y=-14 \\ 7 x-2 y=\quad 2 \end{array}\right.$$

Identifying a Sequence Determine whether the sequence associated with the series is arithmetic or geometric. Find the common difference or ratio and find the sum of the first 15 terms. $$\sum_{n=0}^{\infty} 6(0.8)^{n}$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((5+\sqrt{-16})^{3}\)

Find the sum. $$\sum_{i=2}^{5}\left[(i-1)^{3}+(i+1)^{2}\right]$$

Use Cramer's Rule to solve the system of equations. $$\left\\{\begin{array}{r} -3 x-4 y=-1 \\ 9 x+5 y=-4 \end{array}\right.$$

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((5+\sqrt{-9})^{3}\)

Find the sum. $$\sum_{k=2}^{7}\left[(k+1)+(k-3)^{2}\right]$$

About It In each sequence, decide whether it is possible to fill in the blanks to form an arithmetic sequence. If so, find a recursion formula for the sequence. Explain how you found your answers. (a) -7 \(,\square,\square,\square,\square,\square\) 11 (b) 2\(,\)6\(,\square,\square,\)162 (c) 4,7.5 \(,\square,\square,\square,\square,\square,\) 28.5

A deposit of \(\$ 100\) is made at the beginning of each month in an account that pays \(3 \%\) interest, compounded monthly. The balance \(A\) in the account at the end of 5 years is given by \(A=100\left(1+\frac{0.03}{12}\right)^{1}+\cdots+100\left(1+\frac{0.03}{12}\right)^{60}\) Find \(A\)

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((4+\sqrt{3} i)^{4}\)

Find the sum. $$\sum_{i=0}^{4} 2^{i}$$

Carl Friedrich Gauss, a famous nineteenth century mathematician, was a child prodigy. It was said that when Gauss was \(10,\) he was asked by his teacher to add the numbers from 1 to \(100 .\) Almost immediately, Gauss found the answer by mentally finding the summation. Write an explanation of how he arrived at his conclusion, and then find the formula for the sum of the first \(n\) natural numbers.

A deposit of \(\$ 50\) is made at the beginning of each month in an account that pays \(2 \%\) interest, compounded monthly. The balance \(A\) in the account at the end of 6 years is given by \(A=50\left(1+\frac{0.02}{12}\right)^{1}+\cdots+50\left(1+\frac{0.02}{12}\right)^{72}\) Find \(A.\)

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((5-\sqrt{3} i)^{4}\)

Find the sum. $$\sum_{j=0}^{4}(-2)^{j}$$

A deposit of \(P\) dollars is made at the beginning of each month in an account earning an annual interest rate \(r,\) compounded monthly. The balance \(A\) after \(t\) years is given by $$ \begin{aligned} A=P\left(1+\frac{r}{12}\right) &+P\left(1+\frac{r}{12}\right)^{2}+\cdots \\ &+P\left(1+\frac{r}{12}\right)^{12 t} \end{aligned} $$ Show that the balance is given by \(A=P\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right]\left(1+\frac{12}{r}\right)\)