Chapter 6

The sum of the infinite series \(1+\frac{1}{2}+\frac{1}{2^{2}}+\frac{1}{2^{3}}+\cdots+\frac{1}{2^{n}}\) is \(2 .\) Find values of \(n\) such that \(2-a_{n}<\frac{226-1}{2^{25}}\)

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=\frac{4 n}{3} $$

Find four arithmetic means between 3 and \(18 .\)

In \(15-26,\) write each series in sigma notation. $$ 100+95+90+85+\cdots+5 $$

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ -6+\sum_{n=1}^{8}-6(4)^{n} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=\frac{1}{4}, r=-2 $$

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=\frac{n}{2}+i $$

In \(9-18,\) use the given information to a. write the series in sigma notation, and b. find the sum of the first \(n\) terms. $$ a_{5}=15, d=2, n=12 $$

Find two arithmetic means between 1 and \(5 .\)

In \(15-26,\) write each series in sigma notation. $$ 3+6+9+12+15+\cdots+30 $$

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 1+\sum_{n=1}^{5}(-2)^{n} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=1, r=\sqrt{2} $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 2,4,6,8, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{k=1}^{10} 2 k $$

Write a recursive definition for an arithmetic sequence with a common difference of \(-3\)

In \(15-26,\) write each series in sigma notation. $$ 1+\frac{1}{2}+\frac{1}{4}+\frac{1}{16}+\frac{1}{32} $$

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 1+\sum_{n=1}^{5}\left(\frac{2}{3}\right)^{n} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=10, a_{2}=30 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 3,6,9,12, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{k=2}^{8}(3+k) $$

On July \(1,\) Mr. Taylor owed \(\$ 6,000 .\) On the 1 st of each of the following months, he repaid \(\$ 400\) . List the amount owed by Mr. Taylor on the 2nd of each month starting with July 2 . Explain why the amount owed each month forms an arithmetic sequence.

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}+\frac{6}{7}+\frac{7}{8}+\frac{8}{9}+\frac{9}{10} $$

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 100+\sum_{k=1}^{6} 100\left(\frac{1}{2}\right)^{k} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=-1, a_{2}=4 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 1,4,7,10, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=0}^{9}(20-2 n) $$

Li is developing a fitness program that includes doing push-ups each day. On each day of the first week he did 20 push-ups. Each subsequent week, he increased his daily push-ups by \(5 .\) During which week did he do 60 push-ups a day? a. Use a formula to find the answer to the question. b. Write the arithmetic sequence to answer the question. c. Which method do you think is better? Explain you answer.

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{1 !}+\frac{1}{2 !}+\frac{1}{3 !}+\frac{1}{4 !}+\frac{1}{5 !} $$

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ -81+\sum_{k=1}^{6}-81\left(-\frac{1}{3}\right)^{k} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=100, a_{3}=1 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 3,9,27,81, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{i=0}^{19}(100-5 i) $$

a. Show that a linear function whose domain is the set of positive integers is an arithmetic sequence. b. For the linear function \(y=m x+b, y=a_{n}\) and \(x=n .\) Express \(a_{1}\) and \(d\) of the arithmetic sequence in terms of \(m\) and \(b\) .

In \(15-26,\) write each series in sigma notation. $$ -\frac{1}{3}+\frac{2}{9}-\frac{3}{27}+\frac{4}{81}-\frac{5}{243} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=1, a_{3}=16 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 12,6,3,1.5, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=1}^{25}-2 n $$

Leslie noticed that the daily number of e-mail messages she received over the course of two months form an arithmetic sequence. If she received 13 messages on day 3 and 64 messages on day \(20 :\) a. How many messages did Leslie receive on day 12\(?\) b. How many messages will Leslie receive on day 50\(?\)

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\frac{1}{4 \times 5}+\frac{1}{5 \times 6}+\frac{1}{6 \times 7} $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 7,9,11,13, \dots $$

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=1}^{10}(-1+2 n) $$

A group of students are participating in a math contest. Students receive 1 point for their first correct answer, 2 points for their second correct answer, 4 points for their third correct answer, and so forth. What is the score of a student who answers 10 questions correctly?

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=1, a_{4}=-8 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ 10 i, 8 i, 6 i, 4 i, \ldots $$

Madeline is writing a computer program for class. The first day she wrote 5 lines of code and each day, as she becomes more skilled in writing code, she writes one more line than the previous day. It takes Madeline 6 days to complete the program. How many lines of code did she write?

Heidi deposited \(\$ 400\) at the beginning of each year for six years in an account that paid 5\(\%\) interest. At the end of the sixth year, her first deposit had earned interest for six years and was worth 400\((1.05)^{6}\) dollars, her second deposit had earned interest for five years and was worth 400\((1.05)^{5}\) dollars, her third deposit had earned interest for four years and was worth 400\((1.05)^{4}\) dollars. This pattern continues. a. What is the value of Heidi's sixth deposit at the end of the sixth year? Express your answer as a product and as a dollar value. b. Do the values of these deposits after six years form a geometric sequence? Justify your answer. c. What is the total value of Heidi's six deposits at the end of the sixth year?

In \(15-26,\) write each series in sigma notation. $$ \frac{1}{3}+\frac{2}{9}+\frac{3}{27}+\frac{4}{81}+\frac{5}{243}+\cdots $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=81, a_{5}=1 $$

In \(19-30 :\) a. Write an algebraic expression that represents \(a_{n}\) for each sequence. b. Find the ninth term of each sequence. $$ \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}, \dots $$

Jose is learning to cross-country ski. He began by sking 1 mile the first day and each day he increased the distance skied by 0.2 mile until he reached his goal of 3 miles. a. How many days did it take Jose to reach his goal? b. How many miles did he ski from the time he began until the day he reached his goal?