Solution: Repeating decimalsa. Write the following repeating

Define sequence and give an example.

Suppose the sequence \(\left\{a_{n}\right\}\) is defined by the explicit formula \(a_{n}=1 / n\), for \(n=1,2,3, \ldots\) Write out the first five terms of the sequence.

Suppose the sequence \(\left\{a_{n}\right\}\) is defined by the recurrence relation \(a_{n+1}=n a_{n}\), for \(n=1,2,3, \ldots\) where \(a_{1}=1\). Write out the first five terms of the sequence.

Define finite sum and give an example

Define infinite series and give an example.

Given the series \(\sum_{k=1}^{\infty} k\) ,evaluate the first four terms of its sequence of partial sums \(S_{n}=\sum_{k=1}^{n} k\).

The terms of a sequence of partial sums are defined by \(S_{n}=\sum_{k=1}^{n} k^{2}\) , for \(n=1,2,3, \ldots\). Evaluate the first four terms of the sequence.

Consider the infinite series \(\sum_{k=1}^{\infty} \frac{1}{k}\) . Evaluate the first four terms of the sequence of partial sums.

Explicit formulas Write the first four terms of the sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\). \(a_{n}=1 / 10^{n}\)

Explicit formulas Write the first four terms of the sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\). \(a_{n}=n+1 / n\)

Explicit formulas Write the first four terms of the sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\). \(a_{n}=1+\sin (\pi n / 2)\)

Explicit formulas Write the first four terms of the sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\). \(a_{n}=2 n^{2}-3 n+1\)

Recurrence relations Write the first four terms of the sequence \(\left\{a_{n}\right\}\) defined by the following recurrence relations. \(a_{n+1}=3 a_{n}-12 ; \quad \ a_{1}=10\)

Recurrence relations Write the first four terms of the sequence \(\left\{a_{n}\right\}\) defined by the following recurrence relations. \(a_{n+1}=a_{n}^{2}-1 ; \quad \ a_{1}=1\)

Recurrence relations Write the first four terms of the sequence \(\left\{a_{n}\right\}\) defined by the following recurrence relations. \(a_{n+1}=3 a_{n}^{2}+n+1 ; \quad \ a_{1}=0\)

Recurrence relations Write the first four terms of the sequence \(\left\{a_{n}\right\}\) defined by the following recurrence relations. \(a_{n+1}=a_{n}+a_{n-1} ; \quad \ a_{1}=1, a_{0}=1\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\left\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \ldots\right\}\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\{1,-2,3,-4,5, \ldots\}\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\{1,2,4,8,16, \ldots\}\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\{1,4,9,16,25, \ldots\}\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\{1,3,9,27,81, \ldots\}\)

Enumerated sequences Several terms of a sequence \(\left\{a_{n}\right\}_{n=1}^{\infty}\) are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence. \(\{64,32,16,8,4, \ldots\}\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n}=10^{n}-1 ; \ \ n=1,2,3, \ldots\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n}=n^{8}+1 ; \ \ n=1,2,3, \ldots\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n}=\frac{(-1)^{n}}{n} ; \ \ n=1,2,3, \ldots\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n}=1-10^{-n} ; \ \ n=1,2,3, \ldots\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n+1}=\frac{a_{n}^{2}}{10} ; \ \ a_{0}=1\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n+1}=0.5 a_{n}\left(1-a_{n}\right) ; \ \ a_{0}=0.8\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n+1}=0.5 a_{n}+50 ; \ \ a_{0}=100\)

Limits of sequences Write the terms \(a_{1}, a_{2}, a_{3}\), and \(a_{4}\) of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. \(a_{n+1}=0.9 a_{n}+100 ; \ \ a_{0}=50\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=n+1 ; \ \ n=0,1,2, \ldots\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=2 \tan ^{-1}(1000 n) ; \ \ n=1,2,3, \ldots\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=n^{2}-n ; \ \ n=1,2,3, \ldots\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=\frac{2 n-3}{n} ; \ \ n=1,2,3, \ldots\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=\frac{(n-1)^{2}}{\left(n^{2}-1\right)} ; \ \ n=2,3,4, \ldots\)

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. \(a_{n}=\sin (n \pi / 2) ; \ \ n=0,1,2, \ldots\)

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Convergence or divergence Use a convergence test of your choice to determine whether the following series converge or diverge.

Recurrence relations Write the first four terms of the sequence defined by the following recurrence relations. ; ,

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Enumerated sequences Several terms of a sequence are given. a. Find the next two terms of the sequence, b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence ). c. Find an explicit formula for the general nth term of the sequence.

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Limits of sequences Write the terms , and of the following sequences. If the sequence appears to converge, make a conjecture about its limit. If the sequence diverges, explain why. ;

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist. ;

Explicit formulas for sequences Consider the explicit formulas for the following sequences. a. Find the first four terms of the sequence. b. Using a calculator, make a table with at least 10 terms and determine a plausible value for the limit of the sequence or state that it does not exist.

Practical sequences Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Radioactive decay A material transmutes 50% of its mass to another element every 10 years due to radioactive decay. Let \(M_{n}\) be the mass of the radioactive material at the end of the nth decade, where the initial mass of the material is \(M_{0}=20 \mathrm{\ g}\).

Practical sequences Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Consumer Price Index The Consumer Price Index (the CPI is a measure of the U.S. cost of living) is given a base value of 100 in the year 1984. Assume the CP1 has increased by an average of 3% per year since 1984. Let \(\mathcal{c}_{n}\) be the CPI n years after 1984, where \(c_{0}=100\).

Practical sequences Consider the following situations that generate a sequence. a. Write out the first five terms of the sequence. b. Find an explicit formula for the terms of the sequence. c. Find a recurrence relation that generates the sequence. d. Using a calculator or a graphing utility, estimate the limit of the sequence or state that it does not exist. Drug elimination Jack took a 200-mg dose of a strong pain killer at midnight. Livery hour, 5% of the drug is washed out of his bloodstream. Let \(d_{n}\) be the amount of drug in Jack's blood n hours after the drug was taken, where \(d_{0}=200 \mathrm{\ mg}\).

A square root finder A well-known method for approximating \(\sqrt{c}\) for positive real numbers c consists of the following recurrence relation (based on Newton's method; see Guided Projects). Let \(a_{0}=c\) and \(a_{n+1}=\frac{1}{2}\left(a_{n}+\frac{c}{a_{n}}\right), \quad \ \ \text { for } n=0,1,2,3, \ldots\) a. Use this recurrence relation to approximate \(\sqrt{10}\). How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than 0.01? How many terms of the sequence are needed to approximate \(\sqrt{10}\) with an error less than 0.0001? (To compute the error, assume a calculator gives the exact value.) b. Use this recurrence relation to approximate \(\sqrt{c}\) for c=2, 3, \ldots, 10\). Make a table showing how many terms of the sequence are needed to approximate \(\sqrt{c}\) with an error less than 0.01.

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{3}=0.333 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{6}=0.666 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{1}=0.111 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{5}=0.555 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{09}=0.090909 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{27}=0.272727 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{037}=0.037037 \ldots\)

Repeating decimals a. Write the following repeating decimals as an infinite series. For example, \(0.9999 \ldots=\sum_{k=1}^{\infty} 9\left(0.1^{k}\right)\) b. Find the limit of the sequence of partial sums for the infinite series and express it as a fraction. \(0 . \overline{027}=0.027027 \ldots\)