# 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:

1. Evaluate ∑i=16(3i−1).

2. Evaluate ∑k=115k.

3. 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.