Practice Sigma Notation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

easy

Evaluate

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

Example 2

medium

Re-index

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

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

Example 3

challenge

Evaluate

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

for

n = 4

and identify the general technique.

Example 4

easy

Evaluate

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

Example 5

hard

Use the telescoping identity to evaluate

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

Example 6

easy

Evaluate

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

Example 7

hard

Express

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

in closed form.

Example 8

medium

Evaluate

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

using the shift idea

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

Example 9

medium

Write

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

in sigma notation and evaluate the closed form for

n = 10

Example 10

medium

Does

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

in general?

Example 11

medium

Evaluate

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

Example 12

easy

Evaluate

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

Example 13

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 14

easy

Evaluate

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

Example 15

medium

Rewrite

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

using linearity of summation.

Example 16

medium

Write

3 + 6 + 9 + 12 + 15 + 18

in sigma notation.

Example 17

medium

Use linearity to evaluate

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

Example 18

medium

Evaluate

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

Example 19

challenge

Prove

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

and use it to evaluate

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

Example 20

medium

Evaluate the double sum

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

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Related Concepts

Sequence Series Infinite Geometric Series Riemann Sums