Chapter 12

Find the sum. $$\sum_{k=1}^{3} \frac{1}{k}$$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{2}=0.12, \quad a_{5}=0.00096, \quad n=4 $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=100, d=-5, n=8$$

Find the sum. $$\sum_{j=1}^{100}(-1)^{j}$$

\(43-44\) Simplify using the Binomial Theorem. $$ \frac{(x+h)^{3}-x^{3}}{h} $$

Find the sum. $$ 1+3+9+\cdots+2187 $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a_{1}=55, d=12, n=10$$

Find the sum. $$\sum_{i=1}^{8}\left[1+(-1)^{i}\right]$$

\(43-44\) Simplify using the Binomial Theorem. $$ \frac{(x+h)^{4}-x^{4}}{h} $$

Find the sum. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots-\frac{1}{512} $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a_{2}=8, a_{5}=9.5, n=15$$

Find the sum. $$\sum_{i=4}^{12} 10$$

Show that \((1.01)^{100}>2\) [Hint: Note that \((1.01)^{100}=(1+0.01)^{100}\) and use the Binomial Theorem to show that the sum of the first two terms of the expansion is greater than 2.1

Find the sum. $$ \sum_{k=0}^{10} 3\left(\frac{1}{2}\right)^{k} $$

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$1+5+9+\cdots+401$$

Find the sum. $$\sum_{k=1}^{5} 2^{k-1}$$

Show that \(\left(\begin{array}{l}{n} \\ {0}\end{array}\right)=1\) and \(\left(\begin{array}{l}{n} \\ {n}\end{array}\right)=1\)

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$-3+\left(-\frac{3}{2}\right)+0+\frac{3}{2}+3+\cdots+30$$

Find the sum. $$\sum_{i=1}^{3} i 2^{i}$$

Show that \(\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)=\left(\begin{array}{c}{n} \\ {n-1}\end{array}\right)=n\)

Find the sum of the infinite geometric series. $$ 1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\cdots $$

Use a graphing calculator to evaluate the sum. $$\sum_{k=1}^{10} k^{2}$$

Show that \(\left(\begin{array}{l}{n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{n} \\ {n-r}\end{array}\right) \quad\) for \(0 \leq r \leq n\)

Find the sum of the infinite geometric series. $$ 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\cdots $$

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$-10-9.9-9.8-\dots-0.1$$

Use a graphing calculator to evaluate the sum. $$\sum_{k=1}^{100}(3 k+4)$$

In this exercise we prove the identity $$ \left(\begin{array}{c}{n} \\\ {r-1}\end{array}\right)+\left(\begin{array}{c}{n} \\\ {r}\end{array}\right)=\left(\begin{array}{c}{n+1} \\ {r}\end{array}\right) $$

Find the sum of the infinite geometric series. $$ 1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots $$

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$\sum_{k=0}^{10}(3+0.25 k)$$

Use a graphing calculator to evaluate the sum. $$\sum_{j=7}^{20} j^{2}(1+j)$$

Find the sum of the infinite geometric series. $$ \frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\cdots $$

45–50 A partial sum of an arithmetic sequence is given. Find the sum. $$\sum_{n=0}^{20}(1-2 n)$$

Use a graphing calculator to evaluate the sum. $$\sum_{j=5}^{15} \frac{1}{j^{2}+1}$$

Powers of Factorials Which is larger, \((100 !)^{101}\) or \((101 !)^{100} ?[\text { Hint: Try factoring the expressions. Do they }\) have any common factors?]

Find the sum of the infinite geometric series. $$ \frac{1}{3^{6}}+\frac{1}{3^{8}}+\frac{1}{3^{10}}+\frac{1}{3^{12}}+\cdots $$

Show that a right triangle whose sides are in arithmetic progression is similar to a \(3-4-5\) triangle.

Use a graphing calculator to evaluate the sum. $$\sum_{n=0}^{22}(-1)^{n} 2 n$$

Sums of Binomial Coefficients Add each of the first five rows of Pascal's triangle, as indicated. Do you see a pattern? $$ \begin{array}{c}{1+1=?} \\ {1+2+1=?} \\ {1+3+3+1=?} \\ {1+4+6+4+1=?} \\\ {1+5+10+10+5+1=?}\end{array} $$ Based on the pattern you have found, find the sum of the nth row: $$ \left(\begin{array}{l}{n} \\ {0}\end{array}\right)+\left(\begin{array}{l}{n} \\\ {1}\end{array}\right)+\left(\begin{array}{l}{n} \\\ {2}\end{array}\right)+\cdots+\left(\begin{array}{l}{n} \\\ {n}\end{array}\right) $$ Prove your result by expanding \((1+1)^{n}\) using the Binomial Theorem.

Find the sum of the infinite geometric series. $$ 3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\cdots $$

Find the product of the numbers $$10^{1 / 10}, 10^{2 / 10}, 10^{3 / 10}, 10^{4 / 10}, \ldots, 10^{19 / 10}$$

Use a graphing calculator to evaluate the sum. $$\sum_{n=1}^{100} \frac{(-1)^{n}}{n}$$

Find the sum of the infinite geometric series. $$ -\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots $$

A sequence is harmonic if the reciprocals of the terms of the sequence form an arithmetic sequence. Determine whether the following sequence is harmonic: $$1, \frac{3}{5}, \frac{3}{7}, \frac{1}{3}, \ldots$$

Write the sum without using sigma notation. $$\sum_{k=1}^{5} \sqrt{k}$$

Find the sum of the infinite geometric series. $$ \frac{1}{\sqrt{2}}+\frac{1}{2}+\frac{1}{2 \sqrt{2}}+\frac{1}{4}+\cdots $$

The harmonic mean of two numbers is the reciprocal of the average of the reciprocals of the two numbers. Find the harmonic mean of 3 and \(5 .\)

Write the sum without using sigma notation. $$\sum_{i=0}^{4} \frac{2 i-1}{2 i+1}$$

Express the repeating decimal as a fraction. $$ 0.777 \dots $$

An arithmetic sequence has first term \(a=5\) and common difference \(d=2 .\) How many terms of this sequence must be added to get 2700\(?\)

Write the sum without using sigma notation. $$\sum_{k=0}^{6} \sqrt{k+4}$$