Sigma notation - Sequences and Series

Sigma notation

Don't you find it tiring when we express a series with many terms using numerous addition and/or subtraction signs? Don't you wish that we have something to symbolise this action? Well we have a solution, introducing the "Sigma Notation"! In this section, we will learn how to utilise the sigma notation to represent a series, as well as how to evaluate it.

Basic concepts:
Arithmetic sequences
Arithmetic series
Geometric sequences
Geometric series

Related concepts:
Introduction to sequences
Introduction to infinite series

Lessons

Notes:

$\sum_{i=1}^n$ (an equation containing $i$ )

$\sum$ : "Sigma"; summation of $i^{th}$ term to $n^{th}$ term
$i$ : index, a counter for the $i^{th}$ term
$n$ : index of ending term

1.
Evaluate the following arithmetic series:
3.
Evaluate the following geometric series:
6.
Evaluate the following infinite geometric series:
7.
Write the following sum in sigma notation, then evaluate

Sigma notation

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Lessons

Notes:

∑i=1n\sum_{i=1}^n∑​i=1​n​​(an equation containing iii) ∑\sum∑ : "Sigma"; summation of ithi^{th}i​th​​ term to nthn^{th}n​th​​ term iii : index, a counter for the ithi^{th}i​th​​ term nnn : index of ending term

1.

Evaluate the following arithmetic series:

a)

∑i=110i\sum_{i=1}^{10}i∑​i=1​10​​i

b)

∑i=15−3(i+1)\sum_{i=1}^{5}-3(i+1)∑​i=1​5​​−3(i+1)

2.

Write the following sum in sigma notation, then evaluate 7+9+11+13+...+2057+9+11+13+...+2057+9+11+13+...+205

3.

Evaluate the following geometric series:

a)

∑i=132(12)i\sum_{i=1}^3 2(\frac{1}{2})^i∑​i=1​3​​2(​2​​1​​)​i​​

b)

∑i=120(−3)i+1\sum_{i=1}^{20} (-3)^{i+1}∑​i=1​20​​(−3)​i+1​​

4.

Write the following sum in sigma notation, then evaluate −100+10−1+110−1100-100+10-1+\frac{1}{10}-\frac{1}{100}−100+10−1+​10​​1​​−​100​​1​​

5.

Use sigma notation to express S10S_{10}S​10​​ for −5,10,−20,40,...-5, 10, -20, 40, ...−5,10,−20,40,..., then evaluate

6.

Evaluate the following infinite geometric series:

a)

∑i=1∞3(−5)i−1\sum_{i=1}^{\infty} 3(-5)^{i-1}∑​i=1​∞​​3(−5)​i−1​​

b)

∑i=1∞−72(−23)i\sum_{i=1}^{\infty} -\frac{7}{2}(-\frac{2}{3})^i∑​i=1​∞​​−​2​​7​​(−​3​​2​​)​i​​

7.

Write the following sum in sigma notation, then evaluate

a)

4+2+1+...4+2+1+...4+2+1+...

b)

1−2+4−8+...1-2+4-8+...1−2+4−8+...

Sigma notation

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We have over 1850 practice questions in Algebra for you to master.

Algebra
Arithmetic sequences

Algebra
Arithmetic series

Algebra
Geometric sequences

Algebra
Geometric series

or

$\sum_{i=1}^n$ (an equation containing $i$ )

$\sum$ : "Sigma"; summation of $i^{th}$ term to $n^{th}$ term
$i$ : index, a counter for the $i^{th}$ term
$n$ : index of ending term

$\sum_{i=1}^{10}i$

$\sum_{i=1}^{5}-3(i+1)$

Write the following sum in sigma notation, then evaluate
$7+9+11+13+...+205$

$\sum_{i=1}^3 2(\frac{1}{2})^i$

$\sum_{i=1}^{20} (-3)^{i+1}$

Write the following sum in sigma notation, then evaluate
$-100+10-1+\frac{1}{10}-\frac{1}{100}$

Use sigma notation to express $S_{10}$ for $-5, 10, -20, 40, ...$ , then evaluate

$\sum_{i=1}^{\infty} 3(-5)^{i-1}$

$\sum_{i=1}^{\infty} -\frac{7}{2}(-\frac{2}{3})^i$

$4+2+1+...$

$1-2+4-8+...$