Sigma Notation Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Expand and evaluate โˆ‘k=15(2kโˆ’1)\displaystyle\sum_{k=1}^{5} (2k - 1).

Solution

  1. 1
    Write out each term: k=1:1k=1: 1, k=2:3k=2: 3, k=3:5k=3: 5, k=4:7k=4: 7, k=5:9k=5: 9.
  2. 2
    Sum: 1+3+5+7+9=251+3+5+7+9 = 25.
  3. 3
    Alternatively, use linearity: 2โˆ‘k=15kโˆ’โˆ‘k=151=2โ‹…15โˆ’5=252\sum_{k=1}^5 k - \sum_{k=1}^5 1 = 2 \cdot 15 - 5 = 25.

Answer

2525
Expanding by substituting each value of kk is the most direct approach. The linearity of ฮฃ\Sigma allows splitting the sum and using the formula โˆ‘k=1nk=n(n+1)2\sum_{k=1}^n k = \frac{n(n+1)}{2}.

About Sigma Notation

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.

Learn more about Sigma Notation โ†’

More Sigma Notation Examples

Example 2 medium

Write [formula] in sigma notation and evaluate the closed form for [formula].

Example 3 easy

Evaluate [formula].

Example 4 medium

Rewrite [formula] using linearity of summation.

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