Sigma Notation (Σ) - Summation Formula & Examples
Sigma Notation (Σ) - Summation Formula & Examples
TL;DR
Sigma notation uses the Greek capital letter Σ to write a long sum compactly — for example, ∑i=110i means 1+2+3+⋯+10=55. This article covers how to read sigma notation, the index conventions, the four core summation formulas, three worked examples at increasing difficulty, and the mistakes that quietly cost marks.
What is Sigma Notation?
Sigma notation is a compact way of writing a sum where the terms follow a pattern. The Greek capital letter Σ stands for "sum."
∑i=1nai=a1+a2+a3+⋯+an
Three pieces sit around the sigma:
i=1 below — the starting value of the index (the dummy variable).
n above — the ending value of the index.
ai to the right — the expression to evaluate at each i, then sum.
So ∑i=14i2 reads as: start i at 1, end at 4, square each value of i, add them all up. That is 1+4+9+16=30.
Sigma notation is the difference between writing "1+2+3+⋯+100" and writing ∑i=1100i (where the rule is explicit).
The Sigma Symbol and Its Anatomy
The symbol Σ is the uppercase Greek letter sigma. Eighteenth-century Swiss mathematician Leonhard Euler introduced this convention in his 1755 textbook Institutiones Calculi Differentialis. Before Euler, sums of this kind were written in long-hand with ellipses — clumsy and ambiguous.
| Piece | Meaning |
|---|---|
| Σ | The summation operator — "add up the following" |
| i | The index of summation (also called the dummy variable) |
| i=1 (below) | The lower bound — where the index starts |
| n (above) | The upper bound — where the index ends |
| ai (to the right) | The summand — the rule for each term |
The choice of letter for the index (i, j, k, n — they all work) does not change the value of the sum. ∑i=15i=∑k=15k=15.
How Do You Read Sigma Notation? Three Worked Examples
Quick example
Quick. Evaluate ∑i=15i.
Substitute i=1,2,3,4,5 into the summand and add:
1+2+3+4+5=15
Final answer: 15.
A cross-check using the formula ∑i=1ni=n(n+1)/2: 5⋅6/2=15.
Walking through the wrong answer
Standard. Evaluate ∑i=25(2i+1).
Wrong path. A student who skipped the lower bound writes:
∑i=25(2i+1)=(2⋅1+1)+(2⋅2+1)+⋯+(2⋅5+1)
That answer is wrong. The lower bound says i starts at 2, not 1. The first term should be i=2, giving 2⋅2+1=5.
Correct path. Start i at 2:
∑i=25(2i+1)=(2⋅2+1)+(2⋅3+1)+(2⋅4+1)+(2⋅5+1)=5+7+9+11=32.
Final answer: 32.
Stretch example
Stretch. Evaluate ∑i=120i2.
Working term-by-term takes forever. Use the formula:
∑i=1ni2=n(n+1)(2n+1)/6.
Substitute n=20:
∑i=120i2=20⋅21⋅41/6=2870.
Final answer: 2870.
The Core Summation Formulas
Four formulas cover most school-level sigma problems.
| Sum | Formula |
|---|---|
| Sum of first n natural numbers | ∑i=1ni=n(n+1)/2 |
| Sum of first n squares | ∑i=1ni2=n(n+1)(2n+1)/6 |
| Sum of first n cubes | ∑i=1ni3=[n(n+1)/2]^2 |
| Constant sum | ∑i=1nc=nc |
Properties of Sigma Notation
Five properties that make sigma rearrangeable.
Constant multiple. ∑i=1nc⋅ai=c∑i=1nai. Pull constants out.
Sum rule. ∑i=1n(ai+bi)=∑i=1nai+∑i=1nbi. Split sums.
Difference rule. ∑i=1n(ai−bi)=∑i=1nai−∑i=1nbi.
Index shift. ∑i=1nai=∑j=0n−1aj+1. The dummy variable can be re-labelled.
Splitting the range. ∑i=1nai=∑i=1kai+∑i=k+1nai for any 1≤k<n.
Why Does Sigma Notation Matter?
Sigma notation is not classroom decoration. It is the bridge between arithmetic and calculus.
Riemann sums and integration. The integral ∫abf(x),dx is defined as the limit of a sum ∑f(xi)Δx. Without sigma, you cannot write that definition cleanly.
Statistics — the mean. The sample mean is xˉ=1n∑i=1nxi. Every spreadsheet's
=AVERAGE()is sigma in disguise.Finance — series and annuities. The present value of a stream of payments is ∑i=1nC(1+r)i. Insurance premiums and pension calculations all run on this sum.
Computer science — algorithm complexity. The running time of a nested loop is often ∑i=1ni=n(n+1)/2 — a quadratic in n. Sigma is how a programmer reads the cost of code.
Physics — discrete approximations. Centre of mass, moment of inertia, and total charge in discrete systems are all sigma sums over the constituent particles.
Three Habits That Lose Marks
Mistake 1: Ignoring the lower bound.
Where it slips in: Students default to starting at i=1 even when the notation says otherwise.
Don't do this: ∑i=35i=1+2+3+4+5=15.
The correct way: ∑i=35i=3+4+5=12.
Mistake 2: Treating ∑(ai⋅bi) as (∑ai)⋅(∑bi).
Don't do this: ∑i=13(i⋅i)=(∑i=13i)⋅(∑i=13i)=6⋅6.
The correct way: ∑i=13i2=1+4+9=14.
Mistake 3: Forgetting that the dummy variable is local.
Don't do this: After writing S=∑i=1ni, students set i=n in the result.
The correct way: i does not survive the sum.
The Mathematician Who Shaped The Notation
Leonhard Euler (1707–1783, Switzerland). He standardised the use of Σ for summation in his 1755 textbook Institutiones Calculi Differentialis.
Conclusion
Sigma notation compresses a long sum into a compact expression with four pieces: Σ, the index variable, the bounds, and the summand.
The four core formulas cover most exam problems: sums of 1, i, i2, i3.
Five properties make sigma rearrangeable like algebra.
The most common slip is ignoring the lower bound.
Sigma is the bridge from arithmetic to calculus, statistics, finance, and computer science.
A practical next step
Three problems to practise:
Evaluate ∑i=16(3i−1).
Evaluate ∑k=115k.
Evaluate ∑i=110i2 using the closed-form formula, then verify by summing the first ten squares directly.
Frequently Asked Questions
What is sigma notation?
Sigma notation uses the Greek capital letter Σ to write a long sum compactly.
What does the sigma symbol mean in math?
It means "sum up the following terms."
How do you write a sum in sigma notation?
Identify the pattern of the terms, the index, and its bounds.
What is the formula for the sum of the first n natural numbers?
∑i=1ni=n(n+1)/2.
How do you evaluate a sigma expression?
Either substitute each index value or use a closed-form summation formula.
Can the lower bound be zero or negative?
Yes, any integer value is allowed.
What is the difference between sigma and pi notation?
Σ adds terms together. The Greek capital Π multiplies terms together.