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High School Math : Sequences and Series

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Example Question #1 : Sequences And Series

Evaluate: $1 - \frac{2}{3} + \frac{4}{9} - \frac{8}{27} + ...$

Possible Answers:

None of the other answers are correct.

$\frac{3}{5}$

$\frac{2}{5}$

$\frac{5}{3}$

Correct answer:

$\frac{3}{5}$

Explanation:

This sum can be determined using the formula for the sum of an infinite geometric series, with initial term $a_{0} = 1$ and common ratio $r = -\frac{2}{3}$ :

$S = \frac{a_{0}} {1-r} = \frac{1} {1- \left ( -\frac{2}{3}\right ) } = \frac{1} {1+ \frac{2}{3} }= \frac{1} { \frac{5}{3} } =\frac{3}{5}$

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Example Question #1 : Sequences And Series

The fourth term in an arithmetic sequence is -20, and the eighth term is -10. What is the hundredth term in the sequence?

Possible Answers:

220

105

210

110

Correct answer:

220

Explanation:

An arithmetic sequence is one in which there is a common difference between consecutive terms. For example, the sequence {2, 5, 8, 11} is an arithmetic sequence, because each term can be found by adding three to the term before it.

Let $a_{n}$ denote the nth term of the sequence. Then the following formula can be used for arithmetic sequences in general:

a_n=a_1 +(n-1)d , where d is the common difference between two consecutive terms.

We are given the 4th and 8th terms in the sequence, so we can write the following equations:

a_4=a_1+(4-1)d=a_1+3d=-20

a_8=a_1+(8-1)d=a_1+7d=-10 .

We now have a system of two equations with two unknowns:

a_1+3d=-20

a_1+7d=-10

Let us solve this system by subtracting the equation a_1+7d=-10 from the equation a_1+3d=-20 . The result of this subtraction is

-4d=-10 .

This means that d = 2.5.

Using the equation a_1+7d=-10 , we can find the first term of the sequence.

a_1+7(2.5)=-10

a_1=-27.5

Ultimately, we are asked to find the hundredth term of the sequence.

$a_{100}=a_1+(100-1)d=-27.5+99(2.5)=220$

The answer is 220.

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Example Question #3 : Sequences And Series

Find the sum, if possible:

$3+1+\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+...$

Possible Answers:

$\frac{3}{2}$

$\frac{9}{2}$

Correct answer:

$\frac{9}{2}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms. The key here is finding the rate, or pattern, between the terms. Because this is a geometric sequence, the rate is the constant by which each new term is multiplied.

Plugging in our values, we get:

$S = \frac{3}{1-\frac{1}{3}}$

$S=\frac{3}{\frac{2}{3}}$

$S = \frac{9}{2}$

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Example Question #1 : Sequences And Series

Find the sum, if possible:

$8-6+\frac{9}{2}-\frac{27}{8}+...$

Possible Answers:

$\frac{32}{7}$

$\frac{30}{7}$

$\frac{26}{7}$

$\frac{28}{7}$

$\frac{34}{7}$

Correct answer:

$\frac{32}{7}$

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms in a series

Because the terms switch sign, we know that the rate must be negative.

Plugging in our values, we get:

$S = \frac{8}{1-(\frac{-3}{4})}$

$S = \frac{8}{\frac{7}{4}}$

$S = \frac{32}{7}$

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Example Question #5 : Sequences And Series

Find the sum, if possible:

1-3+9-27+...

Possible Answers:

5267

6725

No solution

7625

2765

Correct answer:

No solution

Explanation:

The formula for the summation of an infinite geometric series is

$S= \frac{a_1}{1-r}$ ,

where a_1 is the first term in the series and is the rate of change between succesive terms in a series.

In order for an infinite geometric series to have a sum, needs to be greater than and less than , i.e. -1<r<1 .

Since r=-3 , there is no solution.

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Example Question #6 : Sequences And Series

Determine the summation notation for the following series:

3+9+27+81+243

Possible Answers:

$\sum_{n=1}^{5}3$

$\sum_{n=1}^{5}n^3$

$\sum_{n=1}^{5}3^{n}$

$\sum_{n=1}^{\infty }n^{3}$

$\sum_{n=1}^{\infty }3^{n}$

Correct answer:

$\sum_{n=1}^{5}3^{n}$

Explanation:

The series is a geometric series. The summation notation of a geometric series is

$\sum_{n=1}^{n}a_{1}\cdot r^{n-1}$ ,

where is the number of terms in the series, $a_{1}$ is the first term of the series, and is the common ratio between terms.

In this series, is , $a_{1}$ is , and is . Therefore, the summation notation of this geometric series is:

$\sum_{n=1}^{5}3\cdot 3^{n-1}$

This simplifies to:

$\sum_{n=1}^{5}3^{n}$

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Example Question #7 : Sequences And Series

Determine the summation notation for the following series:

$10-5+\frac{5}{2}-\frac{5}{4}+\frac{5}{8}$

Possible Answers:

$\sum_{n=1}^{5}20\cdot (-\frac{1}{2})^{n}$

$\sum_{n=1}^{5}-20\cdot (-\frac{1}{2})^{n}$

$\sum_{n=1}^{\infty }-20\cdot (-\frac{1}{2})^{n}$

$\sum_{n=1}^{5}-20\cdot (\frac{1}{2})^{n}$

$\sum_{n=1}^{5}-10\cdot (-\frac{1}{2})^{n}$

Correct answer:

$\sum_{n=1}^{5}-20\cdot (-\frac{1}{2})^{n}$

Explanation:

The series is a geometric series. The summation notation of a geometric series is

$\sum_{n=1}^{n}a_{1}\cdot r^{n-1}$ ,

where is the number of terms in the series, $a_{1}$ is the first term of the series, and is the common ratio between terms.

In this series, is , $a_{1}$ is , and is $-\frac{1}{2}$ . Therefore, the summation notation of this geometric series is:

$\sum_{n=1}^{5}10\cdot (-\frac{1}{2})^{n-1}$

This simplifies to:

$\sum_{n=1}^{5}10\cdot (-\frac{1}{2})^{n}\cdot (-\frac{1}{2})^{-1}$

$\sum_{n=1}^{5}10\cdot (-\frac{1}{2})^{n}\cdot (-2)$

$\sum_{n=1}^{5}-20\cdot (-\frac{1}{2})^{n}$

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Example Question #2 : Sequences And Series

Indicate the sum of the following series:

$\sum_{x=1}^{7}(4x-5)$

Possible Answers:

111

Correct answer:

Explanation:

The formula for the sum of an arithmetic series is

$S = \frac{n}{2}(2(a_1)+(n-1)d)$ ,

where a_1 is the first value in the series, is the number of terms in the series, and is the difference between sequential terms in the series.

In this problem we have:

a_1 = -1

d = 4

n=7

Plugging in our values, we get:

$S = \frac{7}{2}(2(-1)+(6)4)$

S = 3.5(-2+24)

S = 77

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Example Question #9 : Sequences And Series

Indicate the sum of the following series:

$\sum_{c=3}^{10}(8-5c)$

Possible Answers:

Correct answer:

Explanation:

The formula for the sum of an arithmetic series is

$S = \frac{n}{2}(2(a_1)+(n-1)d)$ ,

where a_1 is the first value in the series, is the number of terms in the series, and is the difference between sequential terms in the series.

Here we have:

a_1 = -7

n=8

d = -5

Plugging in our values, we get:

$S = \frac{8}{2}(2(-7)+(8-1)-5)$

S=4(-14-35)

S=-196

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Example Question #10 : Sequences And Series

Indicate the sum of the following series:

$\sum_{m=-2}^{9}(2)^m$

Possible Answers:

1032.25

1230.75

1000.25

1302.75

1023.75

Correct answer:

1023.75

Explanation:

The formula for the sum of a geometric series is

$S=\frac{a_1-a_1r^n}{1-r}$ ,

where a_1 is the first term in the series, is the rate of change between sequential terms, and is the number of terms in the series

For this problem, these values are:

$a_1 = \frac{1}{4}$

r = 2

n = 12

Plugging in our values, we get:

$S=\frac{\frac{1}{4}-\frac{1}{4}(2)^{12}}{1-2}$

S = 1023.75

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High School Math : Sequences and Series

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Sequences And Series

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Example Question #1 : Sequences And Series

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