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Algebra II : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #141 : Mathematical Relationships And Basic Graphs

Evaluate:

$\sum_{k = 1} ^{5} \frac{1}{k}$

Possible Answers:

$2\frac{17}{60}$

$2\frac{19}{60}$

$2\frac{11}{60}$

$2\frac{13}{60}$

$2\frac{71}{60}$

Correct answer:

$2\frac{17}{60}$

Explanation:

$\sum_{k = 1} ^{5} \frac{1}{k}$ is equal to the sum of the expressions formed by substituting 1, 2, 3, 4, and 5, in turn, for in the expression $\frac{1}{k}$ . This is simply the sum of the reciprocals of these 5 integers, which is equal to

$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}+ \frac{1}{5} = 2\frac{17}{60}$

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Example Question #141 : Mathematical Relationships And Basic Graphs

A sequence begins as follows:

3, 4, 6, 9, ...

Which statement is true?

Possible Answers:

The sequence may be arithmetic.

All of these

None of these

The sequence cannot be arithmetic or geometric.

The sequence may be geometric.

Correct answer:

The sequence cannot be arithmetic or geometric.

Explanation:

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the difference between each term and the previous term varies from term to term:

The first difference: 4-3 = 1

The second difference: 6-4 = 2

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratio of each term to the previous one varies from term to term:

The first ratio: $\frac{4}{3}$

The second ratio: $\frac{6}{4}= \frac{3}{2}$

The sequence cannot be geometric.

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Example Question #11 : Other Sequences And Series

A sequence begins as follows:

1,1, 2, 3, 5, 8,...

Which statement is true?

Possible Answers:

The sequence may be geometric.

The sequence cannot be arithmetic or geometric.

None of these

The sequence may be arithmetic.

All of these

Correct answer:

The sequence cannot be arithmetic or geometric.

Explanation:

1-1 = 0

2-1 = 1

The sequence cannot be arithmetic.

$\frac{1}{1} = 1$

$\frac{2}{1}= 2$

The sequence cannot be geometric.

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Example Question #18 : Other Sequences And Series

A sequence begins as follows:

$\frac{1}{5} , \frac{1}{6} , \frac{1}{7} , \frac{1}{8} , ...$

Which statement is true?

Possible Answers:

The sequence may be geometric.

The sequence may be arithmetic.

None of these

The sequence cannot be arithmetic or geometric.

All of these

Correct answer:

The sequence cannot be arithmetic or geometric.

Explanation:

$\frac{1}{6} - \frac{1}{5} = \frac{5}{30} - \frac{6}{30} = -\frac{1}{30}$

$\frac{1}{7} - \frac{1}{6} = \frac{6}{42} - \frac{7}{42} = -\frac{1}{42}$

The sequence cannot be arithmetic.

$\frac{1}{6} \div \frac{1}{5} =\frac{1}{6} \times \frac{5} {1}= \frac{5}{6}$

$\frac{1}{7} \div \frac{1}{6} =\frac{1}{7} \times \frac{6} {1}= \frac{6}{7}$

The sequence cannot be geometric.

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Example Question #2801 : Algebra Ii

A sequence begins as follows:

8, -16, 24, -32, ...

Which statement is true?

Possible Answers:

The sequence may be geometric.

The sequence cannot be arithmetic or geometric.

None of these

The sequence may be arithmetic.

The sequence may be arithmetic and geometric.

Correct answer:

The sequence cannot be arithmetic or geometric.

Explanation:

An arithmetic sequence is one in which each term is generated by adding the same number - the common difference - to the previous term. As can be seen here, the differences between each term and the previous term is not constant from term to term:

-16 - 8 = -24

24 - (-16) = 40

The sequence cannot be arithmetic.

A geometric sequence is one in which each term is generated by multiplying the previous term by the same number - the common ratio. As can be seen here, the ratios of each term to the previous one is not constant from term to term:

$\frac{-16}{8} = -2$

$\frac{24}{-16} = -\frac{3}{2}$

The sequence cannot be geometric.

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Example Question #1 : Factorials

What is the value of 5!+3 .

Possible Answers:

123

None of the other answers.

Correct answer:

123

Explanation:

When evaluating a factorial, you multiply the original number by each integer less than it, stopping at 1.

For this problem, this means that

$5! = 5\cdot 4\cdot 3\cdot 2\cdot 1 = 120$ .

Then adding 3, we get the answer 123

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Example Question #1 : Factorials

Which of the following best represents the approximate value of e^3 ?

Possible Answers:

Correct answer:

Explanation:

The value of is defined as 2.71828 .

To find e^3 , simply cube the decimal number.

$e^3=2.71828\times 2.71828\times 2.71828 = 20.085$

The closest value of this number is:

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Example Question #1 : Factorials

Stewie has $\frac{5!}{3!}$ marbles in a bag. How many marbles does Stewie have?

Possible Answers:

1.66

Correct answer:

Explanation:

5!/3! = (5*4*3*2*1)/(3*2*1)

Simplifying this equation we notice that the 3's, 2's, and 1's cancel so

5!/3! = 5*4 = 20

Alternative Solution

5!/3! = (120/6) = 20

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Example Question #1 : Multiplying And Dividing Factorials

Which of the following is NOT the same as $\frac{6!\times5!}{4!}$ ?

Possible Answers:

$6\times5!$

$6\times5^{2}\times4!$

$\frac{6\times(5!)^2}{4!}$

$5\times6!$

$30\times5!$

Correct answer:

$6\times5!$

Explanation:

The cancels out all of except for the parts higher than 4, this leaves a 6 and a 5 left to multilpy

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Example Question #2 : Multiplying And Dividing Factorials

Simplify the following expression:

$\frac{7!(n+3)!}{6!(n+2)!}$

Possible Answers:

7!(n+3)

$\frac{n+3}{n+2}$

7n +3

7n + 21

n+3

Correct answer:

7n + 21

Explanation:

Recall that (n+3)! = (n+3)(n+2)(n+1)(n)... .

Likewise, $6! = 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1$ .

Thus, the expression $\frac{7!(n+3)!}{6!(n+2)!}$ can be simplified in two parts:

$\frac{7!}{6!} = \frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1} = 7$

and

$\frac{(n+3)!}{(n+2)!} = \frac{(n+3)(n+2)(n+1...)}{(n+2)(n+1)...} = n+3$

The product of these two expressions is the final answer: 7n + 21

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Algebra II : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #141 : Mathematical Relationships And Basic Graphs

Example Question #141 : Mathematical Relationships And Basic Graphs

Example Question #11 : Other Sequences And Series

Example Question #18 : Other Sequences And Series

Example Question #2801 : Algebra Ii

Example Question #1 : Factorials

Example Question #1 : Factorials

Example Question #1 : Factorials

Example Question #1 : Multiplying And Dividing Factorials

Example Question #2 : Multiplying And Dividing Factorials

All Algebra II Resources

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