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High School Math : Pre-Calculus

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Example Questions

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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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Example Question #1 : Using Sigma Notation

Indicate the sum of the following series.

$\sum_{b=2}^{11} \frac{1}{2}(4)^{b-2}$

Possible Answers:

164,772.5

174,762.5

176,742.5

172,764.5

174,726.5

Correct answer:

174,762.5

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

In this problem we have:

$a_1 = \frac{1}{2}$

r = 4

n = 10

Plugging in our values, we get:

$S=\frac{\frac{1}{2}-\frac{1}{2}(4)^{10}}{1-4}$

S = 174,762.5

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Example Question #1 : Terms In A Series

Consider the sequence: $2,\ 5,\ 8,\ 11,\ ...$

What is the fifteenth term in the sequence?

Possible Answers:

Correct answer:

Explanation:

The sequence can be described by the equation $\small 2+3(n-1)$ , where is the term in the sequence.

For the 15th term, n=15 .

$\small 2+3(n-1)=2+3(15-1)$

2+3(14)

2+42

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Example Question #2 : Finding Terms In A Series

What are the first three terms in the series?

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

Possible Answers:

(3)+(7)+(11)

(-1)+(3)+(7)

(1)+(3)+(7)

(-5)+(-1)+(3)

(-1)+(3)+(8)

Correct answer:

(-1)+(3)+(7)

Explanation:

To find the first three terms, replace with , , and .

(4n-5) = (4(1)-5)=-1

(4n-5) = (4(2)-5)=3

(4n-5) = (4(3)-5)=7

The first three terms are , , and .

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Example Question #3 : Finding Terms In A Series

Find the first three terms in the series.

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

Possible Answers:

(3)+(-2)+(-7)

(7)+(12)+(17)

(-7)+(-12)+(-17)

(-7)+(12)+(-17)

(-2)+(-7)+(-12)

Correct answer:

(-7)+(-12)+(-17)

Explanation:

To find the first three terms, replace with , , and .

(8-5n) = (8-5(3))=-7

(8-5n) = (8-5(4))=-12

(8-5n) = (8-5(5))=-17

The first three terms are , , and .

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High School Math : Pre-Calculus

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #5 : Sequences And Series

Example Question #6 : Sequences And Series

Example Question #7 : Sequences And Series

Example Question #2 : Sequences And Series

Example Question #9 : Sequences And Series

Example Question #10 : Sequences And Series

Example Question #1 : Using Sigma Notation

Example Question #1 : Terms In A Series

Example Question #2 : Finding Terms In A Series

Example Question #3 : Finding Terms In A Series

All High School Math Resources

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