Chapter 9

For the following exercises, find the indicated sum. \(\sum_{k=1}^{7} 2^{k}\)

For the following exercises, write the first five terms of the geometric sequence. \(a_{n}=12 \cdot\left(-\frac{1}{2}\right)^{n-1}\)

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{8.9,10.3,11.7, \ldots\\} $$

For the following exercises, write the first eight terms of the sequence. \(a_{1}=2, \quad \mathrm{a}_{2}=10, \quad a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}\)

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum of 3 .

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The seventh term of \((a+b)^{11}\)

, \\#, \\#, @, @, \$, \$, \$, \%, \%, \%, \%\( that begin and end with “… # For the following exercises, find the distinct number of arrangements. The symbols in the string \)\\#, \\#, \\#, @, @, \$, \$, \$, \%, \%, \%, \%$ that begin and end with “\%"

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{-2,-4,-8,-16, \ldots\\}\)

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\\{-0.52,-1.02,-1.52, \ldots\\} $$

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling at least one four or a sum of 8.

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fifth term of \((x-y)^{7}\)

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{1,3,9,27, \ldots\\}\)

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots\right\\} $$

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling an odd sum less than 9.

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The tenth term of \((x-1)^{12}\)

The number of 5 -element subsets from a set containing \(n\) elements is equal to the number of 6 -element subsets from the same set. What is the value of \(n\) ?

For the following exercises, use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. \(-1+3+7+\ldots+31\)

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{-4,-12,-36,-108, \ldots\\}\)

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{-\frac{1}{2},-\frac{5}{4},-2, \ldots\right\\} $$

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum greater than or equal to 15 .

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The ninth term of \(\left(a-3 b^{2}\right)^{11}\)

Can \(C(n, r)\) ever equal \(P(n, r) ?\) Explain.

For the following exercises, use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. \(\sum_{k=1}^{11}\left(\frac{k}{2}-\frac{1}{2}\right)\)

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{0.8,-4,20,-100, \ldots\\}\)

For the following exercises, write a recursive formula for each arithmetic sequence. $$ a=\left\\{\frac{1}{6},-\frac{11}{12},-2, \ldots\right\\} $$

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum less than 15 .

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fourth term of \(\left(x^{3}-\frac{1}{2}\right)^{10}\)

Suppose a set \(A\) has 2,048 subsets. How many distinct objects are contained in \(A\) ?

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\\{-1.25,-5,-20,-80, \ldots\\}\)

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. \(a=\\{7,4,1, \ldots\\} ;\) Find the \(17^{\text {th }}\)

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum less than 6 or greater than 9.

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The eighth term of \(\left(\frac{y}{2}+\frac{2}{x}\right)^{9}\)

How many arrangements can be made from the letters of the word "mountains" if all the vowels must form a string?

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\left\\{-1,-\frac{4}{5},-\frac{16}{25},-\frac{64}{125}, \ldots\right\\}\)

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. \(a=\\{4,11,18, \ldots\\} ;\) Find the \(14^{\text {th }}\)

For the following exercises, evaluate the factorial. \(6 !\)

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum between 6 and 9 , inclusive.

For the following exercises, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^{4}\). Then find and graph each indicated sum on one set of axes. Find and graph \(f_{1}(x),\) such that \(f_{1}(x)\) is the first term of the expansion.

A family consisting of 2 parents and 3 children is to pose for a picture with 2 family members in the front and 3 in the back. (a) How many arrangements are possible with no restrictions? (b) How many arrangements are possible if the parents must sit in the front? (c) How many arrangements are possible if the parents must be next to each other?

For the following exercises, use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(\sum_{k=1}^{9} 2^{k-1}\) \(\begin{array}{ll}-2-10-50-250 \ldots & 0.4-2+10-50 \ldots\end{array}\)

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\left\\{2, \frac{1}{3}, \frac{1}{18}, \frac{1}{108}, \ldots\right\\}\)

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. \(a=\\{2,6,10, \ldots\\} ;\) Find the \(12^{\text {th }}\) \(\begin{array}{lll}\text { term. }\end{array}\)

For the following exercises, evaluate the factorial. $$ \left(\frac{12}{6}\right) ! $$

For the following exercises, two dice are rolled, and the results are summed. Find the probability of rolling a sum of 5 or 6.

For the following exercises, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^{4}\). Then find and graph each indicated sum on one set of axes. Find and graph \(f_{2}(x)\), such that \(f_{2}(x)\) is the sum of the first two terms of the expansion.

A cell phone company offers 6 different voice packages and 8 different data packages. Of those, 3 packages include both voice and data. How many ways are there to choose either voice or data, but not both?

For the following exercises, use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(\sum_{n=1}^{10}-2 \cdot\left(\frac{1}{2}\right)^{n-1}\)

For the following exercises, write an explicit formula for each geometric sequence. \(a_{n}=\left\\{3,-1, \frac{1}{3},-\frac{1}{9}, \ldots\right\\}\)

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence. \(a_{n}=24-4 n\)

For the following exercises, evaluate the factorial. $$ \frac{12 !}{6 !} $$