Summation notation uses the Greek letter sigma, Σ, to represent adding a sequence of terms. Instead of writing every term, the notation specifies a general expression and the starting and ending index. Summation formulas can make long arithmetic calculations shorter and easier to check.
Common summation formulas
- First n positive integers: Σᵢ₌₁ⁿ i = n(n + 1) ÷ 2
- Squares: Σᵢ₌₁ⁿ i² = n(n + 1)(2n + 1) ÷ 6
- Cubes: Σᵢ₌₁ⁿ i³ = [n(n + 1) ÷ 2]²
- Constant: Σᵢ₌₁ⁿ c = nc
- Arithmetic sequence: Sₙ = n(a₁ + aₙ) ÷ 2
For a geometric sequence with first term a₁ and ratio r, the finite sum is Sₙ = a₁(1 − rⁿ) ÷ (1 − r) when r ≠ 1. If r = 1, the sum is na₁.
Worked example
The sum of the integers from 1 to 10 is 10 × 11 ÷ 2 = 55. The formula avoids adding each integer separately. For the squares from 1² to 4², the sum is 4 × 5 × 9 ÷ 6 = 30, matching 1 + 4 + 9 + 16.
How to use sigma notation
- Identify the expression being summed.
- Read the lower index as the starting value and the upper index as the ending value.
- Determine the number of terms and choose a matching formula.
- Substitute the upper limit carefully and simplify.
- For short sums, calculate individual terms to verify the result.
Important details
Check whether the index starts at zero or one; changing the limits changes the terms included. The arithmetic-sequence formula assumes the terms increase or decrease by a constant difference. The geometric formula uses a constant ratio. Other series may require different methods or a recurrence.
Frequently asked questions
What does sigma mean?
It indicates that the specified expression should be added over the listed index values.
Can summation start at zero?
Yes. Use the stated lower and upper limits and include only the terms that fall within them.
Which formula should I use?
Identify the pattern first: consecutive integers, squares, arithmetic progression, or geometric progression each has a specific formula.
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