Sigma notation

Objective. Read, expand and manipulate finite sums.

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2. Worked example
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Learn: the key idea

Sigma notation Σ compactly represents a sum of terms following a pattern, where Σ(from r=1 to n) of f(r) means add f(1)+f(2)+...+f(n). Standard results include Σr=n(n+1)/2 and Σr²=n(n+1)(2n+1)/6, which let you evaluate sums without listing every term. For Σ(from r=1 to 5) of (2r+1), substitute to get 3+5+7+9+11=35, or use the formula Σ(2r+1)=2(n(n+1)/2)+n to confirm. Always check the starting and ending values of r carefully, since shifting limits changes the total.

Worked example

Evaluate the sum given in sigma notation: Σ(from r=1 to 4) of r².

1 $List the terms: 1², 2², 3², 4²$
2 $Compute each term: 1, 4, 9, 16$
3 $Add the terms together: 1+4+9+16$
4 $Simplify to get a total of 30$

Answer: 30

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Common mistake. Students assume Σ(from r=1 to n) of r² equals (Σr)², squaring the sum instead of summing the squares; for n=2 this wrongly gives (1+2)²=9 rather than the correct 1²+2²=5.

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