Using Sigma Notation - High School Math

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Question

Determine the summation notation for the following series:

Answer

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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Question

Determine the summation notation for the following series:

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

Answer

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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Question

Indicate the sum of the following series:

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

Answer

The formula for the sum of an arithmetic series is

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

where 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:

Plugging in our values, we get:

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

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Question

Indicate the sum of the following series:

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

Answer

The formula for the sum of an arithmetic series is

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

where 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:

Plugging in our values, we get:

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

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Question

Indicate the sum of the following series:

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

Answer

The formula for the sum of a geometric series is

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

where 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:

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

Plugging in our values, we get:

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

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Question

Indicate the sum of the following series.

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

Answer

The formula for the sum of a geometric series is

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

where 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:

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

Plugging in our values, we get:

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

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