Chapter 12

Find the sum of each series. $$\sum_{i=1}^{19}(-3-4 k)$$

Use any or all of the methods described in this section to solve each problem. From 10 names on a ballot, 4 will be elected to a political party committee. In how many ways can the committee of 4 be formed if each person will have a different responsibility?

Find each sum. $$\sum_{i=1}^{\infty}\left(-\frac{1}{3}\right)\left(\frac{3}{4}\right)^{i-1}$$

Find the sum for each series. $$\sum_{i=1}^{6}\left(2+i-i^{2}\right)$$

Find the sum of each series. $$\sum_{i=1}^{1000} i$$

Use any or all of the methods described in this section to solve each problem. The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} 2^{k} \leq 62$$

Find the sum for each series. $$\sum_{i=1}^{4}\left(3 i^{3}+2 i-4\right)$$

Use any or all of the methods described in this section to solve each problem. A typical combination for a padlock consists of 3 numbers from 0 to 39 . Find the number of combinations that are possible with this type of lock if a number may be repeated. (Hint: The word combination is a misnomer. Lock combinations are permutations because the arrangement of the numbers is important.)

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} 3(0.5)^{k} \leq 2.8$$

Find the sum for each series. $$\sum_{i=1}^{6}\left(i^{2}+2 i^{3}\right)$$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. $$a_{n}=4.2 n+9.73$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{i=1}^{n} \frac{4}{2^{k}}<4$$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. $$a_{n}=8.42 n+36.18$$

Use any or all of the methods described in this section to solve each problem. To win the jackpot in a lottery game, a person must pick 3 numbers from 0 to 9 in the correct order. If a number can be repeated, how many ways are there to play the game?

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} \frac{2}{5^{k}}<\frac{1}{2}$$

Find the sum for each series. $$\sum_{i=1}^{43}\left(15 i^{2}-2\right)$$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth. $$a_{n}=\sqrt{8} n+\sqrt{3}$$

Use any or all of the methods described in this section to solve each problem. How many distinguishable ways can 4 keys be put on a circular key ring? (Hint: Consider that clockwise and counterclockwise arrangements are not different.)

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth. $$a_{n}=-\sqrt[3]{4} n+\sqrt{7}$$

Use any or all of the methods described in this section to solve each problem. How many ways can 7 people sit at a round table? Assume "a different way" means that at least 1 person is sitting next to someone different.

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{j=1}^{6}-(3.6)^{j}$$

Find the sum for each series. $$\sum_{i=1}^{52}\left(i^{2}+27 i+180\right)$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n} 2 k \leq 42$$

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, n-1)=P(n, n)$$

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{j=3}^{8} 2(0.4)^{j}$$

Use summation notation to write each series. Start the index at \(i=1\) $$\frac{2}{5(1)}+\frac{2}{5(2)}+\frac{2}{5(3)}+\dots+\frac{2}{5(100)}$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{k=1}^{n}(k+2) \leq 52$$

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, 1)=n$$

Use a graphing calculator to evaluate each sum. Round to the nearest thousandth. $$\sum_{i=4}^{9} 3(0.25)^{i}$$

Use summation notation to write each series. Start the index at \(i=1\) $$\frac{1}{1+1}+\frac{2}{2+1}+\frac{3}{3+1}+\dots+\frac{25}{25+1}$$

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{i=1}^{n}(2 k-1)<26$$

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$P(n, 0)=1$$

Find the future value of each annuity. Payments of \(\$ 1000\) at the end of each year for 9 years at \(2 \%\) interest compounded annually

Use summation notation to write each series. Start the index at \(i=1\) $$1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\dots+\frac{1}{9}$$

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, n)=1$$

Find the future value of each annuity. Payments of \(\$ 800\) at the end of each year for 12 years at \(1 \%\) interest compounded annually

Use summation notation to write each series. Start the index at \(i=1\) $$-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\frac{1}{81}-\dots-\frac{1}{2187}$$

Find the sum of all the integers from 51 to 71.

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, 0)=1$$

Find the future value of each annuity. Payments of \(\$ 2430\) at the end of each year for 10 years at \(2.5 \%\) interest compounded annually

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=\frac{n+4}{2 n}$$

Find the sum of all the integers from \(-8\) to 30.

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, 1)=n$$

Solve each problem. Payments of \(\$ 1500\) at the end of each year for 6 years at \(1.5 \%\) interest compounded annually

Use a graphing calculator to graph the first 10 terms of each sequence. Make a conjecture as to whether the sequence comerges or diverges. If you think it converges, make a conjecture about the mumber to which it converges. $$a_{n}=\frac{1+4 n}{2 n}$$

If a clock strikes the proper number of chimes each hour on the hour, how many times will it chime in a month of 30 days?

Prove each statement for positive integers \(n\) and \(r\), with \(r \leq n\). (Hint: Use the definitions of permutations and combinations.) $$C(n, n-1)=n$$

Solve each problem. MODELING Investment for Retirement According to T. Rowe Price Associates, a person who has a moderate investment strategy and \(n\) years until retirement should have accumulated savings of \(a_{n}\) percent of his or her annual salary. The geometric sequence $$ a_{n}=1276(0.916)^{n} $$ gives the appropriate percent for each year \(n\) (a) Find \(a_{1}\) and \(r\) (b) Find and interpret the terms \(a_{10}\) and \(a_{20}\)

A stack of telephone poles has 30 in the bottom row, 29 in the next, and so on, with one pole in the top row. How many poles are in the stack?