Sigma Notation Examples: 48 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Sigma Notation.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Sigma notation uses the Greek letter Σ to express the sum of many terms compactly. The expression $\sum_{i=1}^{n} a_i$ means 'add up $a_i$ for every integer $i$ from 1 to $n$ .' For example, $\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30$ .

Sigma notation is shorthand for 'add these all up.' The letter below $\Sigma$ is a counter, the number below is where to start, the number above is where to stop, and the expression to the right tells you what to add each time.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Sigma packs a long sum into a start, a stop, a counter, and a rule for each term.

Common stuck point: The procedure for sigma notation is the easy part; the trap is treating the term rule as a constant and multiplying by the number of terms. Asking "Is this an instruction to add up terms generated by substituting an index over a range?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is this an instruction to add up terms generated by substituting an index over a range?

Worked Examples

Example 1

easy

Expand and evaluate

\displaystyle\sum_{k=1}^{5} (2k - 1)

Answer

25

First step

Write out each term:

k=1: 1

k=2: 3

k=3: 5

k=4: 7

k=5: 9

Full solution

2
Sum: $1+3+5+7+9 = 25$ .
3
Alternatively, use linearity: $2\sum_{k=1}^5 k - \sum_{k=1}^5 1 = 2 \cdot 15 - 5 = 25$ .

Expanding by substituting each value of

k

is the most direct approach. The linearity of

\Sigma

allows splitting the sum and using the formula

\sum_{k=1}^n k = \frac{n(n+1)}{2}

Example 2

medium

Write

1^2 + 2^2 + 3^2 + \cdots + n^2

in sigma notation and evaluate the closed form for

n = 10

Example 3

medium

Use the closed form to evaluate

\sum_{i=1}^{20} i

Example 4

medium

Use linearity to evaluate

\sum_{i=1}^{10} (4i + 3)

Example 5

hard

Evaluate

\sum_{i=1}^{n} i^3

for

n=5

using the closed form.

Example 6

hard

Use a closed form to simplify

\sum_{i=1}^{n}(2i-1)

Example 7

medium

Shift the index: rewrite

\sum_{i=3}^{8} (i-2)^2

so it starts at

j=1

Example 8

medium

Show

\sum_{i=1}^{n} c\cdot a_i = c\sum_{i=1}^{n} a_i

by expanding for

n=3

Example 9

hard

Use the telescoping identity to evaluate

\sum_{i=1}^{n} \left(\frac{1}{i} - \frac{1}{i+1}\right)

Example 10

challenge

Evaluate

\sum_{i=1}^{100} (3i - 2)

using closed forms.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Evaluate

\displaystyle\sum_{i=0}^{4} 3^i

Example 2

medium

Rewrite

\displaystyle\sum_{j=1}^{n}(3j^2 + 2j - 1)

using linearity of summation.

Example 3

easy

Evaluate

\sum_{i=1}^{4} i

Example 4

easy

Evaluate

\sum_{i=1}^{3} i^2

Example 5

easy

How many terms are in

\sum_{i=0}^{5} a_i

Example 6

easy

Evaluate

\sum_{i=1}^{4} 3

Example 7

easy

Evaluate

\sum_{i=2}^{4} (2i)

Example 8

easy

Use

\sum_{i=1}^{n} i = \frac{n(n+1)}{2}

to evaluate

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

Example 9

easy

Rewrite

2 + 4 + 6 + 8 + 10

in sigma notation.

Example 10

easy

Evaluate

\sum_{k=1}^{3} 2^k

Example 11

medium

Evaluate

\sum_{i=1}^{5} (2i + 1)

Example 12

medium

Evaluate

\sum_{i=1}^{4} (i^2 - i)

Example 13

medium

Evaluate

\sum_{i=3}^{6} i

using the shift idea

\sum_{i=1}^{6} i - \sum_{i=1}^{2} i

Example 14

medium

Evaluate

\sum_{i=1}^{6} i^2

using

\frac{n(n+1)(2n+1)}{6}

Example 15

medium

True or false:

\sum_{i=1}^{n} (a_i b_i) = \left(\sum a_i\right)\left(\sum b_i\right)

. Justify with

a_i = b_i = i

n=2

Example 16

medium

Evaluate

\sum_{i=1}^{4} 5 \cdot 2^{i-1}

Example 17

medium

Rewrite

\sum_{i=1}^{n} (3i - 2)

as a single closed-form expression in

n

Example 18

medium

Evaluate the double sum

\sum_{i=1}^{2}\sum_{j=1}^{3} (i + j)

Example 19

medium

Re-index

\sum_{i=1}^{5} (i+2)^2

as a sum of squares with shifted bounds, then evaluate.

Example 20

challenge

Find a closed form for

\sum_{i=1}^{n} \frac{1}{i(i+1)}

and evaluate at

n=99

Example 21

challenge

Evaluate

\sum_{i=1}^{n} i \cdot 2^{i}

for

n = 4

and identify the general technique.

Example 22

challenge

Prove

\sum_{i=1}^{n} (2i - 1) = n^2

and use it to evaluate

\sum_{i=1}^{50}(2i-1)

Example 23

easy

Evaluate

\sum_{i=1}^{5} i

Example 24

easy

Evaluate

\sum_{i=1}^{4} i^2

Example 25

easy

Evaluate

\sum_{i=1}^{5} 7

Example 26

medium

Evaluate

\sum_{k=1}^{5} (k^2 - k)

Example 27

medium

Write

3 + 6 + 9 + 12 + 15 + 18

in sigma notation.

Example 28

medium

Write

1 + 1/2 + 1/4 + 1/8 + 1/16

in sigma notation.

Example 29

medium

Evaluate

\sum_{i=1}^{6} 2^i

Example 30

medium

Evaluate

\sum_{i=1}^{8} (-1)^i

Example 31

hard

Use

\sum i = n(n+1)/2

and

\sum i^2 = n(n+1)(2n+1)/6

to evaluate

\sum_{i=1}^{n} i(i+1)

for

n=6

Example 32

easy

\sum_{i=1}^{n} a_i

i

is called the ___.

Example 33

medium

Evaluate

\sum_{i=1}^{4} (i+1)(i-1)

Example 34

medium

What is the closed-form value of

\sum_{i=1}^{n} 1

Example 35

medium

Evaluate

\sum_{i=1}^{4} \frac{1}{i(i+1)}

Example 36

easy

Evaluate

\sum_{i=2}^{5} (i^2 - 1)

Example 37

hard

Express

\sum_{i=1}^{n} (2i+1)

in closed form.

Example 38

medium

Evaluate

\sum_{i=0}^{3} (i^2 + 1)

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Related Concepts

Sequence Series Infinite Geometric Series Riemann Sums

Background Knowledge

These ideas may be useful before you work through the harder examples.

sequenceseries