Chapter 9

, \\#, \\#, @, \$, \%, \%, \%, \%, \%\( # For the following exercises, find the distinct number of arrangements. The symbols in the string \)\\#, \\#, \\#, @, \$, \%, \%, \%, \%, \%$

Write the first eight terms of the sequence. $$a_{1}=2, a_{2}=10, a_{n}=\frac{2\left(a_{n-1}+2\right)}{a_{n-2}}$$

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

Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. $$ -1.7+-0.4+0.9+2.2+3.5+4.8 $$

,*,#,@,@,\$,\$,\$,\%,\%,\%,\% 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, find the indicated term of each binomial without fully expanding the binomial. The seventh term of \((a+b)^{11}\)

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. $$ a=\\{-0.52,-1.02,-1.52, \ldots\\} $$

Write a recursive formula for each sequence. $$-2.5,-5,-10,-20,-40, \dots$$

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

Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. $$ 6+\frac{15}{2}+9+\frac{21}{2}+12+\frac{27}{2}+15 $$

The set, \(S\) consists of 900,000,000 whole numbers, each being the same number of digits long. How many digits long is a number from S? (Hint: use the fact that a whole number cannot start with the digit 0.)

For the following exercises, find the indicated term of each binomial without fully expanding the binomial. The fi h 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. $$ a=\left\\{\frac{1}{5}, \frac{9}{20}, \frac{7}{10}, \ldots\right\\} $$

Write a recursive formula for each sequence. $$-8,-6,-3,1,6, \dots$$

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

Use the formula for the sum of the first \(n\) terms of an arithmetic series to find the sum. $$ -1+3+7+\ldots+31 $$

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 \(n\) ? (Hint: the order in which the element for the subsets are chosen is not important.)

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

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. $$ a=\left\\{-\frac{1}{2},-\frac{5}{4},-2, \ldots\right\\} $$

Write a recursive formula for each sequence. $$2,4,12,48,240, \dots$$

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 .

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

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

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

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. $$ a=\left\\{\frac{1}{6},-\frac{11}{12},-2, \ldots\right\\} $$

Write a recursive formula for each sequence. $$35,38,41,44,47, \dots$$

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

Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(S_{6}\) for the series \(-2-10-50-250 \ldots\)

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

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

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 each sequence. $$ 15,3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \ldots $$

Write a recursive formula for each sequence. $$15,3, \frac{3}{5}, \frac{3}{25}, \frac{3}{125}, \dots$$

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 \(9 .\)

Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. \(S_{7}\) for the series \(0.4-2+10-50 \ldots\)

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

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, evaluate the factorial. $$ 6 ! $$

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.

Use the formula for the sum of the first \(n\) terms of a geometric series to find the partial sum. $$ \sum_{k=1}^{9} 2^{k-1} $$

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 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 fi st term of the expansion.

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