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High School Math : Using Sigma Notation

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Example Questions

Example Question #31 : Pre Calculus

Determine the summation notation for the following series:

3+9+27+81+243

Possible Answers:

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

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

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

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

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

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 #1 : Sequences And Series

Determine the summation notation for the following series:

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

Possible Answers:

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

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

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

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

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

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 #1 : 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 #1 : 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 #2 : Using Sigma Notation

Indicate the sum of the following series:

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

Possible Answers:

1230.75

1302.75

1000.25

1023.75

1032.25

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:

172,764.5

174,726.5

176,742.5

164,772.5

174,762.5

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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High School Math : Using Sigma Notation

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #31 : Pre Calculus

Example Question #1 : Sequences And Series

Example Question #1 : Sequences And Series

Example Question #1 : Sequences And Series

Example Question #2 : Using Sigma Notation

Example Question #1 : Using Sigma Notation

All High School Math Resources

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