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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

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Example Questions

Example Question #333 : Solving Equations

Solve for :

$\frac{a+5}{7}-\frac{a-8}{2}=5$

Possible Answers:

$a=\frac{11}{5}$

$a=\frac{9}{5}$

$a=-\frac{9}{11}$

$a=-\frac{4}{5}$

$a=-\frac{5}{11}$

Correct answer:

$a=-\frac{4}{5}$

Explanation:

Step 1: The least common denominator of 7 and 2 is 14. Multiply each term by this LCD:

$14\left (\frac{a+5}{7} \right )-14\left ( \frac{a-8}{2} \right )=5\left ( 14 \right )$

Step 2: Reduce the fractions:

$2a+10-\left ( 7a-56 \right )=70$

Step 3: Isolate :

2a+10-7a+56=70

Remember to distribute the negative:

-5a+66=70

-5a=4

${\color{Red} a=-\frac{4}{5}}$

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Example Question #821 : Basic Single Variable Algebra

Solve for and .

4x+7y=-72

8x-4y=0

Possible Answers:

$x=2\text{ and }y=9$

$x=-7\text{ and }y=-2$

$x=3\text{ and }y=5$

$x=12\text{ and }y=15$

$x=-4\text{ and }y=-8$

Correct answer:

$x=-4\text{ and }y=-8$

Explanation:

Solve for x and y by using substitution. Solve the first equation for x.

$x=\frac{-72-7y}{4}$

Next substitute x into the second equation and simplify,

$8(\frac{-72-7y}{4})+-4y=0$

-144-14y-4y=0

-18y=144

y=-8

Now use the value for y to solve for x.

$x=\frac{-72-7(-8)}{4}=-4$

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Example Question #1441 : High School Math

Which of the following is a geometric sequence?

Possible Answers:

$t_{0}=4$

$t_{n}=t_{n-1}+3.4$

$t_{0}=2$

$t_{n} = t_{n-1}^{6}$

$t_{0}=10$

$t_{n}=t_{n-1}+5$

$t_{0}=5$

$t_{n}=t_{n-1}+3$

$t_{0}=4$

$t_{n}=t_{n-1}\times 3$

Correct answer:

$t_{0}=4$

$t_{n}=t_{n-1}\times 3$

Explanation:

A geometric sequence is one in which the next term is found by mutlplying the previous term by a particular constant. Thus, we look for an implicit definition which involves multiplication of the previous term. The only possibility is:

$t_{0}=4$

$t_{n}=t_{n-1}\times 3$

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Example Question #1 : Geometric Sequences

$\left \{ -1, 2, -4, 8, -16, 32... \right \}$

What is the explicit formula for the above sequence? What is the 20th value?

Possible Answers:

$a_{n}=a_{1}\cdot-2;\ a_{20}=-524288$

$a_{n}= -1\cdot -2^{n-1};\ a_{20} = 524288$

$a_{n}=a_{n-1}\cdot-2^{n-1};\ a_{20} = -262144$

$a_{n}=a_{n-1}\cdot-1;\ a_{20}=524288$

$a_{n-1}= a_{n}\cdot -2;\ a_{20}=-524288$

Correct answer:

$a_{n}= -1\cdot -2^{n-1};\ a_{20} = 524288$

Explanation:

This is a geometric series. The explicit formula for any geometric series is:

$a_{n}= a_{1} \cdot r^{n-1}$ , where is the common ratio and is the number of terms.

In this instance r = -2 and a_1=-1 .

$a_{n}= a_{1} \cdot r^{n-1}\rightarrow a_n=(-1)\cdot(-2)^{n-1}$

Substitute n = 20 into the equation to find the 20th term:

$a_{20}=(-1)\cdot(-2)^{20-1}$

$a_{20}=-(-2)^{19}=-(-524288)=524288$

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Example Question #1 : Summations And Sequences

What type of sequence is shown below?

$\left \{ 4,9,15,22,30... \right \}$

Possible Answers:

Multiplicative

Arithmetic

Subtractive

Geometric

None of the other answers

Correct answer:

None of the other answers

Explanation:

This series is neither geometric nor arithmetic.

A geometric sequences is multiplied by a common ratio () each term. An arithmetic series adds the same additional amount () to each term. This series does neither.

Mutiplicative and subtractive are not types of sequences.

Therefore, the answer is none of the other answers.

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Example Question #1 : Mathematical Relationships And Basic Graphs

Identify the 10th term in the series:

$\left \{ \frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32}... \right \}$

Possible Answers:

$\frac{1}{1634}$

$\frac{1}{512}$

$\frac{1}{8192}$

$\frac{1}{4096}$

$\frac{1}{2048}$

Correct answer:

$\frac{1}{2048}$

Explanation:

The explicit formula for a geometric series is $a_{n}=a_{1}\cdot r^{n-1}$

In this problem $r=\frac{1}{2}$

Therefore:

$a_{10}=\frac{1}{4}\cdot \frac{1}{2}^{9}$

${\color{Red} a_{10}=\frac{1}{2048}}$

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Example Question #1 : Geometric Sequences

Which of the following could be the formula for a geometric sequence?

Possible Answers:

a_4=a_3+9(n+1)

$a_4=5\cdot (\frac{1}{3})^4$

$a_{16}=a_{15}\cdot 3$

a_3=6+4(n-1)

$a_6=3\cdot 7^5, a_1=3$

Correct answer:

$a_6=3\cdot 7^5, a_1=3$

Explanation:

The explicit formula for a geometric series is $a_n=a_1\cdot r^{n-1}$ .

Therefore, ${\color{Red} a_6=3\cdot 7^5, a_1=3}$ is the only answer that works.

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Example Question #4 : Summations And Sequences

Find the 15th term of the following series:

$\left \{ 120, 60, 30, 15, 7.5,... \right \}$

Possible Answers:

$a_{15}=.0025$

$a_{15}=.0018$

$a_{15}=.0073$

$a_{15}=.0037$

$a_{15}=.0016$

Correct answer:

$a_{15}=.0073$

Explanation:

This series is geometric. The explicit formula for any geometric series is:

$a_n=a_1\cdot r^{n-1}$

Where represents the $n^{th}$ term, a_1 is the first term, and is the common ratio.

In this series $r = \frac{1}{2}$ .

Therefore the formula to find the 15th term is:

$a_{15}=120\cdot \left(\frac{1}{2}\right)^{14}$

$a_{15}=.0073$

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Example Question #2 : How To Find The Missing Number In A Set

2, 6, 18, n, 162...

$\textup{In the above geometric sequence, what is the value of }n\textup{?}$

Possible Answers:

Correct answer:

Explanation:

$\textup{Look for a pattern. Each term in the sequence is three times the previous term.}$

$18\ast3=54$

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Example Question #7 : Mathematical Relationships And Basic Graphs

Give the 33rd term of the Geometric Series

2, -3, 4.5, -6.75, ...

[2 is the first term]

Possible Answers:

431,439. 884

-1,294,319.650

1,853,020,188, 851, 841

862,879.767

12,987

Correct answer:

862,879.767

Explanation:

First we need to find the common ratio by dividing the second term by the first:

$-3 \div 2 = -1.5$

The $n^{th}$ term is

$\\a_{n} = a_1(r)^{n-1}\\a_n=2(-1.5)^{n-1}$ ,

so the 33rd term will be

$a_{33} = 2(-1.5)^ {32 } = 862,879.767$ .

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Algebra II : Algebra II

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #333 : Solving Equations

Example Question #821 : Basic Single Variable Algebra

Example Question #1441 : High School Math

Example Question #1 : Geometric Sequences

Example Question #1 : Summations And Sequences

Example Question #1 : Mathematical Relationships And Basic Graphs

Example Question #1 : Geometric Sequences

Example Question #4 : Summations And Sequences

Example Question #2 : How To Find The Missing Number In A Set

Example Question #7 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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