FoundationsConcept reference

Product notation

A finite indexed product combines the values of a stated rule at each integer index in an inclusive range by multiplication.

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On this page 7 sections

Overview
Indexed product notation tells you which values to multiply
The lower limit need not be 1
Include zero factors: one zero makes the product zero
Key takeaway
Sources & further reading
Concept connections

Figure 1Indexed product of k+1 for k=1,2,3: factors 2, 3 and 4 multiply to 24. Three factor boxes are joined by multiplication signs.

01Indexed product notation tells you which values to multiply#

Indexed product notation tells you which values to multiply. The capital pi symbol is followed by a rule for each factor.

Read the starting and ending indices.

\prod_{k=1}^{3}(k+1)

In $\prod_{k=1}^{3}(k+1)$ , the index $k$ takes the integer values $1,2,3$ , including both ends. Substitute each value into the rule.

Write one factor per index.

k+1:\quad2,3,4

The product $\prod_{k=1}^{3}(k+1)$ includes the factors $2,3,4$ . Multiply them. The index moves through its range; it is not one fixed unknown to solve for.

Evaluate the expansion.

2\cdot3\cdot4=24

Check your reasoning

Evaluate $\prod_{k=2}^{3}(k+1)$ .

Show answer and explanation

The factors are $3$ and $4$ , so $3\cdot4=12$ .

02The lower limit need not be 1#

The lower limit need not be $1$ . The index letter is only a local label: changing $k$ to $i$ consistently changes nothing. If the rule is constant, include one copy per index.

Three copies of a constant.

\prod_{i=1}^{3}2=2\cdot2\cdot2=8

Check your reasoning

A log has $a_1=3$ , $a_2=2$ , $a_3=4$ . Evaluate $\prod_{i=1}^3 a_i$ .

Show answer and explanation

Read the listed entries in the given range, then multiply: $3\cdot2\cdot4=24$ .

03Include zero factors: one zero makes the product zero#

Include zero factors: one zero makes the product zero.

Check your reasoning

For $a_1=3,a_2=0,a_3=5$ , evaluate $\prod_{i=1}^3a_i$ .

Show answer and explanation

$3\cdot0\cdot5=0$ .

Key takeaway

List the integer indices from the lower through the upper limit. Substitute each into the rule, then multiply the resulting values.

Expand and evaluate a finite indexed product

Sources & further reading

[1]
University of Western Australia, Summation and Product Notation, Chapter 10 ↗University of Western Australia · Book

01Indexed product notation tells you which values to multiply#

02The lower limit need not be 1#

03Include zero factors: one zero makes the product zero#

List the integer indices from the lower through the upper limit. Substitute each into the rule, then multiply the resulting values.

Sources & further reading

Reference this concept