High School Math : Using Sigma Notation

Study concepts, example questions & explanations for High School Math

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Example Questions

Example Question #2 : Sequences And Series

Determine the summation notation for the following series:

\(\displaystyle 3+9+27+81+243\)

Possible Answers:

\(\displaystyle \sum_{n=1}^{5}3^{n}\)

\(\displaystyle \sum_{n=1}^{5}3\)

\(\displaystyle \sum_{n=1}^{5}n^3\)

\(\displaystyle \sum_{n=1}^{\infty }n^{3}\)

\(\displaystyle \sum_{n=1}^{\infty }3^{n}\)

Correct answer:

\(\displaystyle \sum_{n=1}^{5}3^{n}\)

Explanation:

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

\(\displaystyle \sum_{n=1}^{n}a_{1}\cdot r^{n-1}\),

where \(\displaystyle n\) is the number of terms in the series, \(\displaystyle a_{1}\) is the first term of the series, and \(\displaystyle r\) is the common ratio between terms.

In this series, \(\displaystyle n\) is \(\displaystyle 5\)\(\displaystyle a_{1}\) is \(\displaystyle 3\), and \(\displaystyle r\) is \(\displaystyle 3\). Therefore, the summation notation of this geometric series is:

\(\displaystyle \sum_{n=1}^{5}3\cdot 3^{n-1}\)

This simplifies to:

\(\displaystyle \sum_{n=1}^{5}3^{n}\)

Example Question #1 : Sequences And Series

Determine the summation notation for the following series:

\(\displaystyle 10-5+\frac{5}{2}-\frac{5}{4}+\frac{5}{8}\)

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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

\(\displaystyle \sum_{n=1}^{n}a_{1}\cdot r^{n-1}\),

where \(\displaystyle n\) is the number of terms in the series, \(\displaystyle a_{1}\) is the first term of the series, and \(\displaystyle r\) is the common ratio between terms.

In this series, \(\displaystyle n\) is \(\displaystyle 5\)\(\displaystyle a_{1}\) is \(\displaystyle 10\), and \(\displaystyle r\) is \(\displaystyle -\frac{1}{2}\). Therefore, the summation notation of this geometric series is:

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

This simplifies to:

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

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

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

Example Question #1 : Sequences And Series

Indicate the sum of the following series:

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

Possible Answers:

\(\displaystyle 99\)

\(\displaystyle 77\)

\(\displaystyle 66\)

\(\displaystyle 111\)

\(\displaystyle 88\)

Correct answer:

\(\displaystyle 77\)

Explanation:

The formula for the sum of an arithmetic series is

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

where \(\displaystyle a_1\) is the first value in the series, \(\displaystyle n\) is the number of terms in the series, and \(\displaystyle d\) is the difference between sequential terms in the series.

In this problem we have:

\(\displaystyle a_1 = -1\)

\(\displaystyle d = 4\)

\(\displaystyle n=7\)

Plugging in our values, we get:

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

\(\displaystyle S = 3.5(-2+24)\)

\(\displaystyle S = 77\)

Example Question #1 : Sequences And Series

Indicate the sum of the following series:

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

Possible Answers:

\(\displaystyle -196\)

\(\displaystyle -200\)

\(\displaystyle -192\)

\(\displaystyle -204\)

\(\displaystyle -208\)

Correct answer:

\(\displaystyle -196\)

Explanation:

The formula for the sum of an arithmetic series is

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

where \(\displaystyle a_1\) is the first value in the series, \(\displaystyle n\) is the number of terms in the series, and \(\displaystyle d\) is the difference between sequential terms in the series.

Here we have:

\(\displaystyle a_1 = -7\)

\(\displaystyle n=8\)

\(\displaystyle d = -5\)

Plugging in our values, we get:

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

\(\displaystyle S=4(-14-35)\)

\(\displaystyle S=-196\)

Example Question #2 : Using Sigma Notation

Indicate the sum of the following series:

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

Possible Answers:

\(\displaystyle 1230.75\)

\(\displaystyle 1302.75\)

\(\displaystyle 1000.25\)

\(\displaystyle 1023.75\)

\(\displaystyle 1032.25\)

Correct answer:

\(\displaystyle 1023.75\)

Explanation:

The formula for the sum of a geometric series is

\(\displaystyle S=\frac{a_1-a_1r^n}{1-r}\),

where \(\displaystyle a_1\) is the first term in the series, \(\displaystyle r\) is the rate of change between sequential terms, and \(\displaystyle n\) is the number of terms in the series

For this problem, these values are:

\(\displaystyle a_1 = \frac{1}{4}\)

\(\displaystyle r = 2\)

\(\displaystyle n = 12\)

Plugging in our values, we get:

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

\(\displaystyle S = 1023.75\)

Example Question #1 : Using Sigma Notation

Indicate the sum of the following series.

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

Possible Answers:

\(\displaystyle 172,764.5\)

\(\displaystyle 174,726.5\)

\(\displaystyle 176,742.5\)

\(\displaystyle 164,772.5\)

\(\displaystyle 174,762.5\)

Correct answer:

\(\displaystyle 174,762.5\)

Explanation:

The formula for the sum of a geometric series is

\(\displaystyle S=\frac{a_1-a_1r^n}{1-r}\),

where \(\displaystyle a_1\) is the first term in the series, \(\displaystyle r\) is the rate of change between sequential terms, and \(\displaystyle n\) is the number of terms in the series

In this problem we have:

\(\displaystyle a_1 = \frac{1}{2}\)

\(\displaystyle r = 4\)

\(\displaystyle n = 10\)

Plugging in our values, we get:

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

\(\displaystyle S = 174,762.5\)

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