Chapter 13

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ \sum_{k=0}^{10}(3+0.25 k) $$

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1-\frac{1}{3}+\frac{1}{9}-\frac{1}{27}+\cdots $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{n=0}^{22}(-1)^{n} 2 n $$

\(49-54\) . A partial sum of an arithmetic sequence is given. Find the sum. $$ \sum_{n=0}^{20}(1-2 n) $$

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ \frac{2}{5}+\frac{4}{25}+\frac{8}{125}+\cdots $$

\(49-54\) . Use a graphing calculator to evaluate the sum. $$ \sum_{n=1}^{100} \frac{(-1)^{n}}{n} $$

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

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1+\frac{3}{2}+\left(\frac{3}{2}\right)^{2}+\left(\frac{3}{2}\right)^{3}+\cdots $$

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=1}^{5} \sqrt{k} $$

Difference in Volumes of Cubes The volume of a cube of side \(x\) inches is given by \(V(x)=x^{3},\) so the volume of a cube of side \(X+2\) inches is given by \(V(x+2)=(x+2)^{3}\) . Use the Binomial Theorem to show that the difference in volume between the larger and smaller cubes is \(6 x^{2}+12 x+8\) cubic inches

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

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ \frac{1}{3^{6}}+\frac{1}{3^{8}}+\frac{1}{3^{10}}+\frac{1}{3^{12}}+\cdots $$

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{i=0}^{4} \frac{2 i-1}{2 i+1} $$

Probability of Hitting a Target The probability that an archer hits the target is \(p=0.9,\) so the probability that he misses the target is \(q=0.1 .\) It is known that in this situation the probability that the archer hits the target exactly \(r\) times in \(n\) attempts is given by the term containing \(p^{r}\) in the binomial expansion of \((p+q)^{n} .\) Find the probability the archer hits the target exactly three times in five attempts.

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

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+\cdots $$

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=0}^{6} \sqrt{k+4} $$

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

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 \(5 .\)

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 1-1+1-1+\cdots $$

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=6}^{9} k(k+3) $$

Sums of Binomial Coefficients Add each of the first five rows of Pascal’s triangle, as indicated. Do you see a pattern? $$ \begin{array}{c}{1+1=?} \\ {1+2+1=?} \\ {1+3+3+1=?} \\ {1+4+6+4+1=?} \\\ {1+5+10+10+5+1=?}\end{array} $$ On the basis of 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.

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\(?\)

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ 3-3(1.1)+3(1.1)^{2}-3(1.1)^{3}+\cdots $$

\(55-60\) . Write the sum without using sigma notation. $$ \sum_{k=3}^{100} X^{k} $$

An arithmetic sequence has first term \(a_{1}=1\) and fourth term \(a_{4}=16 .\) How many terms of this sequence must be added to get 2356\(?\)

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ -\frac{100}{9}+\frac{10}{3}-1+\frac{3}{10}-\cdots $$

Depreciation The purchase value of an office computer is \(\$ 12,500 .\) Its annual depreciation is \(\$ 1875 .\) Find the value of the computer after 6 years.

Determine whether the infinite geometric series is convergent or divergent. If it is convergent, find its sum. $$ \frac{1}{\sqrt{2}}+\frac{1}{2}+\frac{1}{2 \sqrt{2}}+\frac{1}{4}+\cdots $$

\(61-68\) Write the sum using sigma notation. $$ 1+2+3+4+\dots+100 $$

Poles in a Pile Telephone poles are being stored in a pile with 25 poles in the first layer, 24 in the second, and so on. If there are 12 layers, how many telephone poles does the pile contain?

\(61-68\) Write the sum using sigma notation. $$ 2+4+6+\dots+20 $$

Salary Increases A man gets a job with a salary of \(\$ 30,000\) a year. He is promised a \(\$ 2300\) raise each subsequent year. Find his total earnings for a 10 -year period.

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

\(61-68\) Write the sum using sigma notation. $$ 1^{2}+2^{2}+3^{2}+\cdots+10^{2} $$

Drive-In Theater \(A\) drive-in theater has spaces for 20 cars in the first parking row, 22 in the second, 24 in the third, and so on. If there are 21 rows in the theater, find the number of cars that can be parked.

Express the repeating decimal as a fraction. $$ 0.2 \overline{53} $$

\(61-68\) Write the sum using sigma notation. $$ \frac{1}{2 \ln 2}-\frac{1}{3 \ln 3}+\frac{1}{4 \ln 4}-\frac{1}{5 \ln 5}+\cdots+\frac{1}{100 \ln 100} $$

Theater Seating An architect designs a theater with 15 seats in the first row, 18 in the second, 21 in the third, and so on. If the theater is to have a seating capacity of \(870,\) how many rows must the architect use in his design?

Express the repeating decimal as a fraction. $$ 0.030303 \ldots $$

\(61-68\) Write the sum using sigma notation. $$ \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\cdots+\frac{1}{999 \cdot 1000} $$

Falling Ball When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls 16 \(\mathrm{ft}\) in the first second, 48 \(\mathrm{ft}\) in the next second, 80 \(\mathrm{ft}\) in the next second, and so on. (a) Find the total distance a ball falls in 6 \(\mathrm{s}\) . (b) Find a formula for the total distance a ball falls in \(n\) seconds.

Express the repeating decimal as a fraction. $$ 2.11 \overline{25} $$

\(61-68\) Write the sum using sigma notation. $$ \frac{\sqrt{1}}{1^{2}}+\frac{\sqrt{2}}{2^{2}}+\frac{\sqrt{3}}{3^{2}}+\cdots+\frac{\sqrt{n}}{n^{2}} $$

The Twelve Days of Christmas In the well-known song "The Twelve Days of Christmas," a person gives his sweetheart \(k\) gifts on the \(k\) th day for each of the 12 days of Christmas. The person also repeats each gift identically on each subsequent day. Thus, on the 12 th day the sweetheart receives a gift for the first day, 2 gifts for the second, 3 gifts for the third, and so on. Show that the number of gifts received on the 12 th day is a partial sum of an arithmetic sequence. Find this sum.

Express the repeating decimal as a fraction. $$ 0 . \overline{112} $$

\(61-68\) Write the sum using sigma notation. $$ 1+x+x^{2}+x^{3}+\dots+x^{100} $$

Arithmetic Means The arithmetic mean (or average) of two numbers \(a\) and \(b\) is $$ m=\frac{a+b}{2} $$ (a) Insert two arithmetic means between 10 and 18 . (b) Insert three arithmetic means between 10 and 18 . (c) Suppose a doctor needs to increase a patient's dosage of a certain medicine from 100 \(\mathrm{mg}\) to 300 \(\mathrm{mg}\) per day in five equal steps. How many arithmetic means must be inserted between 100 and 300 to give the progression of daily doses, and what are these means?

Express the repeating decimal as a fraction. $$ 0.123123123 \ldots $$

\(61-68\) Write the sum using sigma notation. $$ 1-2 x+3 x^{2}-4 x^{3}+5 x^{4}+\cdots-100 x^{99} $$