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 ฮฃ to express the sum of many terms compactly. The expression โˆ‘i=1nai\sum_{i=1}^{n} a_i means 'add up aia_i for every integer ii from 1 to nn.' For example, โˆ‘i=14i2=1+4+9+16=30\sum_{i=1}^{4} i^2 = 1 + 4 + 9 + 16 = 30.

Sigma notation is shorthand for 'add these all up.' The letter below ฮฃ\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 โˆ‘i=14i2\sum_{i=1}^{4} i^2.

Example 2

medium
Re-index โˆ‘i=15(i+2)2\sum_{i=1}^{5} (i+2)^2 as a sum of squares with shifted bounds, then evaluate.

Example 3

challenge
Evaluate โˆ‘i=1niโ‹…2i\sum_{i=1}^{n} i \cdot 2^{i} for n=4n = 4 and identify the general technique.

Example 4

easy
Evaluate โˆ‘k=132k\sum_{k=1}^{3} 2^k.

Example 5

hard
Use the telescoping identity to evaluate โˆ‘i=1n(1iโˆ’1i+1)\sum_{i=1}^{n} \left(\frac{1}{i} - \frac{1}{i+1}\right).

Example 6

easy
Evaluate โˆ‘i=157\sum_{i=1}^{5} 7.

Example 7

hard
Express โˆ‘i=1n(2i+1)\sum_{i=1}^{n} (2i+1) in closed form.

Example 8

medium
Evaluate โˆ‘i=36i\sum_{i=3}^{6} i using the shift idea โˆ‘i=16iโˆ’โˆ‘i=12i\sum_{i=1}^{6} i - \sum_{i=1}^{2} i.

Example 9

medium
Write 12+22+32+โ‹ฏ+n21^2 + 2^2 + 3^2 + \cdots + n^2 in sigma notation and evaluate the closed form for n=10n = 10.

Example 10

medium
Does โˆ‘i=1n(aiโ‹…bi)=(โˆ‘ai)(โˆ‘bi)\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 โˆ‘i=141i(i+1)\sum_{i=1}^{4} \frac{1}{i(i+1)}.

Example 12

easy
Evaluate โˆ‘i=14i\sum_{i=1}^{4} i.

Example 13

medium
True or false: โˆ‘i=1n(aibi)=(โˆ‘ai)(โˆ‘bi)\sum_{i=1}^{n} (a_i b_i) = \left(\sum a_i\right)\left(\sum b_i\right). Justify with ai=bi=ia_i = b_i = i, n=2n=2.

Example 14

easy
Evaluate โˆ‘i=25(i2โˆ’1)\sum_{i=2}^{5} (i^2 - 1).

Example 15

medium
Rewrite โˆ‘j=1n(3j2+2jโˆ’1)\displaystyle\sum_{j=1}^{n}(3j^2 + 2j - 1) using linearity of summation.

Example 16

medium
Write 3+6+9+12+15+183 + 6 + 9 + 12 + 15 + 18 in sigma notation.

Example 17

medium
Use linearity to evaluate โˆ‘i=110(4i+3)\sum_{i=1}^{10} (4i + 3).

Example 18

medium
Evaluate โˆ‘i=162i\sum_{i=1}^{6} 2^i.

Example 19

challenge
Prove โˆ‘i=1n(2iโˆ’1)=n2\sum_{i=1}^{n} (2i - 1) = n^2 and use it to evaluate โˆ‘i=150(2iโˆ’1)\sum_{i=1}^{50}(2i-1).

Example 20

medium
Evaluate the double sum โˆ‘i=12โˆ‘j=13(i+j)\sum_{i=1}^{2}\sum_{j=1}^{3} (i + j).
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