Chapter 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. \(\frac{1}{2 !}+\frac{1}{3 !}+\cdots+\frac{1}{(n+1) !}+\cdots\)

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

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ 3,6,9,12, \ldots, n=8 $$

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

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ 1+5+25+\cdots+a_{n}, n=10 $$

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

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ 2,7,12,17, \ldots, n=12 $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=5}^{15}[4 n-(n+1)] $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots+a_{n}, n=6 $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. \(1.11111 \ldots=1+0.1+0.01+0.001+\cdots\)

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. $$ 1,-10,100,-1,000,10,000, \ldots $$

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

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. $$ a_{1}=24, a_{n}=0, d=-6 $$

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ 18,16,14,12, \ldots, n=10 $$

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

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ 2-8+32-\cdots+a_{n}, n=7 $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. \(0.33333 \ldots=0.3+0.03+0.003+0.0003+\cdots\)

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. $$ 1,0.1,0.01,0.001,0.0001, \ldots $$

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=2 n-1 $$

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ \frac{1}{2}, 1, \frac{3}{2}, 2, \ldots, n=7 $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{k=3}^{5}(5-4 k) $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=4, a_{5}=324, n=9 $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. 0.444444\(\ldots\)

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. $$ 0.05,-0.1,0.2,-0.4, \dots $$

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

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. $$ a_{1}=2, d=\frac{1}{2}, n=15 $$

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ -1,-3,-5,-7, \ldots, n=10 $$

In \(3-14 :\) a. Write each arithmetic series as the sum of terms. b. Find each sum. $$ \sum_{n=0}^{5}(-2 n)^{n+1} $$

In \(3-14,\) find the sum of \(n\) terms of each geometric series. $$ a_{1}=1, a_{8}=128, n=10 $$

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. $$ a, a^{2}, a^{3}, a^{4}, \dots $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. \(0.121212 \ldots=0.12+0.0012+0.000012+\cdots\)

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

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. $$ a_{1}=0, d=-2, n=10 $$

In \(9-14 :\) a. Find the common difference of each arithmetic sequence. b. Write the \(n\) th term of each sequence for the given value of \(n .\) $$ 2.1,2.2,2.3,2.4, \ldots, n=20 $$

In \(15-26,\) write each series in sigma notation. $$ 3+5+7+9+11+13+15 $$

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

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=1, r=6 $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. 0.242424\(\dots\)

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

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. $$ a_{1}=\frac{1}{3}, d=\frac{1}{3}, n=12 $$

Write the first six terms of the arithmetic sequence that has 12 for the first term and 42 for the sixth term.

In \(15-26,\) write each series in sigma notation. $$ 1+6+11+16+21+26+31+36 $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=40, r=\frac{1}{2} $$

An infinitely repeating decimal is an infinite geometric series. Find the rational number represented by each of the following infinitely repeating decimals. 0.126126126\(\dots\)

In \(3-18,\) write the first five terms of each sequence. $$ a_{n}=12-3 n $$

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. $$ a_{1}=100, d=-5, n=20 $$

Write the first nine terms of the arithmetic sequence that has 100 as the fifth term and 80 as the ninth term.

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

In \(15-22 :\) a. Write each sum as a series. b. Find the sum of each series. $$ 10+\sum_{n=1}^{5} 10\left(\frac{1}{2}\right)^{n} $$

In \(15-26,\) write the first five terms of each geometric sequence. $$ a_{1}=2, r=3 $$