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Summation notation

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

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On this page 7 sections

Overview
Indexed sum notation tells you which values to add
The lower limit need not be 1
Keep each term’s sign when adding the expansion
Key takeaway
Sources & further reading
Concept connections

Figure 1Indexed sum of k+2 for k=1,2,3: adjacent lengths 3,4,5 form a total length of 12. All widths use the same unit.

01Indexed sum notation tells you which values to add#

Indexed sum notation tells you which values to add. The capital sigma symbol is followed by a rule for each term.

Read the starting and ending indices.

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

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

Write one term per index.

k+2:\quad3,4,5

The sum $\sum_{k=1}^{3}(k+2)$ includes the values $3,4,5$ . Add them. The index moves through its range; it is not one fixed unknown to solve for.

Evaluate the expansion.

3+4+5=12

Check your reasoning

Evaluate $\sum_{k=2}^{4}(k+1)$ .

Show answer and explanation

The terms are $3,4,5$ , so $3+4+5=12$ .

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.

\sum_{i=1}^{3}2=2+2+2=6

Check your reasoning

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

Show answer and explanation

Read the listed entries in the given range, then add: $3+2+4=9$ .

03Keep each term’s sign when adding the expansion#

Keep each term’s sign when adding the expansion.

Check your reasoning

For $a_1=3,a_2=-2,a_3=5$ , evaluate $\sum_{i=1}^3a_i$ .

Show answer and explanation

$3+(-2)+5=6$ .

Key takeaway

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

Expand and evaluate a finite indexed sum

Sources & further reading

[1]
OpenStax College Algebra 2e, 9.4 ↗OpenStax · Book

01Indexed sum notation tells you which values to add#

02The lower limit need not be 1#

03Keep each term’s sign when adding the expansion#

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

Sources & further reading

Reference this concept