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Algebra II : Summations and Sequences

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Example Question #1 : Geometric Sequences

Which of the following is a geometric sequence?

Possible Answers:

$t_{0}=5$

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

$t_{0}=2$

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

$t_{0}=10$

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

$t_{0}=4$

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

$t_{0}=4$

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

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 #2 : Mathematical Relationships And Basic Graphs

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

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

Possible Answers:

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

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

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

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

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

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 #3 : Mathematical Relationships And Basic Graphs

What type of sequence is shown below?

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

Possible Answers:

None of the other answers

Geometric

Multiplicative

Arithmetic

Subtractive

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 : Geometric Sequences

Identify the 10th term in the series:

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

Possible Answers:

$\frac{1}{4096}$

$\frac{1}{2048}$

$\frac{1}{8192}$

$\frac{1}{512}$

$\frac{1}{1634}$

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:

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

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

a_4=a_3+9(n+1)

a_3=6+4(n-1)

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

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 #6 : Mathematical Relationships And Basic Graphs

Find the 15th term of the following series:

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

Possible Answers:

$a_{15}=.0018$

$a_{15}=.0025$

$a_{15}=.0073$

$a_{15}=.0016$

$a_{15}=.0037$

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 #1 : 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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Example Question #8 : Mathematical Relationships And Basic Graphs

Find the 19th term of the sequence $7,000, \enspace 2,800, \enspace 1,120, \enspace 448, ...$

[the first term is 7,000]

Possible Answers:

14,551, 915.228

0.000481

1.119

0.000192

0.00000000184

Correct answer:

0.000481

Explanation:

First find the common ratio by dividing the second term by the first:

$2,800 \div 7,000 = \frac{ 2}{5} = 0.4$

Since the first term is 7,000 , the nth term can be found using the formula

$\\a_{n} = a_1(r)^{n-1}$

$7,000(0.4)^ {n-1 }$ ,

so the 19th term is $7,000(0.4)^{18} = 0.000481$

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Example Question #9 : Mathematical Relationships And Basic Graphs

Find the 21st term of the sequence 90, 108, 129.6, 155.52, ...

[90 is 1st, so n=1]

Possible Answers:

2.348

55.206

3,450.384

1.956

4,140.461

Correct answer:

3,450.384

Explanation:

First, find the common ratio by dividing the second term by the first:

$108 \div 90 = 1.2$

The nth term can be found using

$\\a_{n} = a_1(r)^{n-1}$

$90(1.2)^ {n-1}$ ,

so the 21st term is

$90(1.2) ^ {20 } = 3,450.384$ .

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Algebra II : Summations and Sequences

Study concepts, example questions & explanations for Algebra II

All Algebra II Resources

Example Questions

Example Question #1 : Geometric Sequences

Example Question #2 : Mathematical Relationships And Basic Graphs

Example Question #3 : Mathematical Relationships And Basic Graphs

Example Question #1 : Geometric Sequences

Example Question #1 : Geometric Sequences

Example Question #6 : Mathematical Relationships And Basic Graphs

Example Question #1 : How To Find The Missing Number In A Set

Example Question #7 : Mathematical Relationships And Basic Graphs

Example Question #8 : Mathematical Relationships And Basic Graphs

Example Question #9 : Mathematical Relationships And Basic Graphs

All Algebra II Resources

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