Chapter 6

Casey said that the formula for the sum of a geometric series could be written as \(S_{n}=\frac{a_{1}-a_{n} r}{1-r} .\) Do you agree with Casey? Justify your answer.

Show that if the first term of an infinite geometric series is 1 and the common ratio is \(\frac{1}{c},\) then the sum is \(\frac{c}{c-1} .\)

Is there more than one arithmetic series such that the sum of the first and the last terms is 80 and the sum of the terms is \(1,200 ?\) Justify your answer.

Nichelle said that sequence of numbers in which each term equals half of the previous term is a finite sequence. Randi said that is an infinite sequence. Who is correct? Justify your answer.

Explain why \(\sum_{k=0}^{10} \frac{1}{k}\) is undefined.

Cody said that since the calculator gives the value of \(e\) as \(2.71828,\) the value of \(e\) can be written as \(2.71828,\) a repeating decimal and therefore a rational number. Do you agree with Cody? Explain why or why not.

Sierra said that \(8,8 \sqrt{2}, 16,16 \sqrt{2}, 32\) is a gcometric sequence with three geometric means, \(8 \sqrt{2}, 16,\) and 16\(\vee 2 .\) Do you agree with Sierra? Justify your answer.

Is \(1+1+2+3+5+8+13+21\) an arithmetic series? Justify your answer.

Pedro said that to form a sequence of five terms that begins with 2 and ends with \(12,\) you should divide the difference between 12 and 2 by 5 to find the common difference. Do you agree with Pedro? Explain why or why not.

a. Jacob said that if \(a_{n}=3 n-1\) , then \(a_{n+1}=a_{n}+3 .\) Do you agree with Jacob? Explain why or why not. b. Carlos said that if \(a_{n}=2^{n}\) , then \(a_{n+1}=2^{n+1} .\) Do you agree with Carlos? Explain why or why not.

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=1}^{10} 3 n $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=1, r=2, n=12 $$

a. Write each series in sigma notation. b. Determine whether each sum increases without limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that limit. \(1+\frac{1}{3}+\frac{1}{9}+\cdots\)

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 4,8,16,32,64, \dots $$

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ 2+4+6+8+10+12 $$

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 2,5,8,11,14, \dots $$

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

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{5}(2 k-2) $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=4, r=3, n=11 $$

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 1,5,25,125,625, \ldots $$

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ 10+20+30+40+50+60 $$

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ -3 i,-1 i, 1 i, 3 i, 5 i, \dots $$

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=n+5 $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{4} k^{2} $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{2}=6, r=4, n=15 $$

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

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 1,1,2,3,5,8, \dots $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=1}^{6} n^{3} $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=10, r=10, n=6 $$

a. Write each series in sigma notation. b. Determine whether each sum increases without limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that limit. \(5+1+\frac{1}{5}+\frac{1}{25}+\cdots\)

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ \frac{1}{2}, 2,8,32, \ldots $$

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ 0 i+4 i+8 i+12 i+16 i+20 i $$

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 20,15,10,5,0, \dots $$

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=\frac{1}{n} $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=1}^{10}(100-5 k) $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{3}=0.4, r=2, n=12 $$

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 1,-3,9,-27,81, \dots $$

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ 0+\frac{1}{3}+\frac{2}{3}+1+\frac{4}{3}+\frac{5}{3}+2 $$

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

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 1,2,4,8,16, \dots $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=5}^{10}(3 n-3) $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=1, r=\frac{1}{3}, n=10 $$

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 36,12,4, \frac{4}{3}, \dots $$

a. Write each series in sigma notation. b. Determine whether each sum increases without limit, decreases without limit, or approaches a finite limit. If the series has a finite limit, find that limit. \(6+3+\frac{3}{2}+\frac{3}{4}+\cdots\)

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=20-n $$

In \(3-8,\) find the sum of each series using the formula for the partial sum of an arithmetic series. Be sure to show your work. $$ \sqrt{2}+2 \sqrt{2}+3 \sqrt{2}+4 \sqrt{2}+\cdots+15 \sqrt{2} $$

In \(3-8,\) determine if each sequence is an arithmetic sequence. If the sequence is arithmetic, find the common difference. $$ 1,1.25,1.5,1.75,2, \dots $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=2}^{5}\left(n^{2}+2 i\right) $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ 5+10+20+\cdots+a_{n}, n=8 $$

In \(3-14,\) determine whether each given sequence is geometric. If it is geometric, find \(r\) . If it is not geometric, explain why it is not. $$ 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots $$