Sigma Notation (Σ) - Summation Formula & Examples

What is Sigma Notation?

Sigma notation is a compact way of writing a sum where the terms follow a pattern. The Greek capital letter $\Sigma$ stands for "sum."

$$\sum_{i=1}^{n} a_i = a_1 + a_2 + a_3 + \dots + a_n$$

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.

• $a_i$ to the right — the expression to evaluate at each $i$, then sum.

So $\sum_{i=1}^{4} i^2$ 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 "$1 + 2 + 3 + \dots + 100$" (and hoping the reader fills in the gap) and writing $\sum_{i=1}^{100} i$ (where the rule is explicit).

The Sigma Symbol and Its Anatomy

The symbol $\Sigma$ 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.

The choice of letter for the index ($i$, $j$, $k$, $n$ — they all work) does not change the value of the sum. $\sum_{i=1}^{5} i = \sum_{k=1}^{5} k = 15$.

How Do You Read Sigma Notation? Three Worked Examples

We will work through three problems — Quick, Standard, and Stretch. The Standard one starts with the most common wrong path.

Quick example

Quick. Evaluate $\sum_{i=1}^{5} i$.

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 $\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}$: $\dfrac{5 \cdot 6}{2} = 15$. Agreement.

Walking through the wrong answer

Standard. Evaluate $\sum_{i=2}^{5} (2i + 1)$.

Wrong path. A student who skipped the lower bound writes:

$$\sum_{i=2}^{5}(2i + 1) = (2 \cdot 1 + 1) + (2 \cdot 2 + 1) + \dots + (2 \cdot 5 + 1) = 3 + 5 + 7 + 9 + 11 = 35$$

That answer is wrong. The lower bound says $i$ starts at $2$, not $1$. The first term should be $i = 2$, giving $2 \cdot 2 + 1 = 5$.

Correct path. Start $i$ at $2$:

$$\sum_{i=2}^{5} (2i + 1) = (2 \cdot 2 + 1) + (2 \cdot 3 + 1) + (2 \cdot 4 + 1) + (2 \cdot 5 + 1) = 5 + 7 + 9 + 11 = 32$$

Final answer: $32$.

In Bhanzu's Grade 10 cohorts, the "started at 1 by reflex" slip shows up on roughly four out of ten first attempts when the lower bound isn't $1$. A Bhanzu trainer who sees this draws a vertical line through the lower-bound number with a red marker before the student starts — a 5-second habit that locks the index attention on the right value.

Stretch example

Stretch. Evaluate $\sum_{i=1}^{20} i^2$.

Working term-by-term takes forever. Use the formula:

$$\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$

Substitute $n = 20$:

$$\sum_{i=1}^{20} i^2 = \frac{20 \cdot 21 \cdot 41}{6} = \frac{17220}{6} = 2870$$

Final answer: $2870$.

A sanity check: the sum has $20$ terms, the largest is $400$, the smallest is $1$. The average must lie somewhere between, and $2870 / 20 = 143.5$ — that is in range.

The Core Summation Formulas

Four formulas cover most school-level sigma problems.

The cubes formula has a neat side-effect: the sum of the first $n$ cubes equals the square of the sum of the first $n$ natural numbers. Fibonacci's contemporary Nicomachus noticed this around 100 CE.

Properties of Sigma Notation

Five properties that make sigma rearrangeable.

• Constant multiple. $\sum_{i=1}^{n} c \cdot a_i = c \sum_{i=1}^{n} a_i$. Pull constants out.

• Sum rule. $\sum_{i=1}^{n} (a_i + b_i) = \sum_{i=1}^{n} a_i + \sum_{i=1}^{n} b_i$. Split sums.

• Difference rule. $\sum_{i=1}^{n} (a_i - b_i) = \sum_{i=1}^{n} a_i - \sum_{i=1}^{n} b_i$.

• Index shift. $\sum_{i=1}^{n} a_i = \sum_{j=0}^{n-1} a_{j+1}$. The dummy variable can be re-labelled.

• Splitting the range. $\sum_{i=1}^{n} a_i = \sum_{i=1}^{k} a_i + \sum_{i=k+1}^{n} a_i$ for any $1 \leq k < n$.

These five properties are the reason sigma notation is useful — sums can be rearranged like ordinary algebra, just on a more compact symbol.

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 $\int_a^b f(x) , dx$ is defined as the limit of a sum $\sum f(x_i) \Delta x$. Without sigma, you cannot write that definition cleanly.

• Statistics — the mean. The sample mean is $\bar{x} = \dfrac{1}{n} \sum_{i=1}^{n} x_i$. Every spreadsheet's =AVERAGE() is sigma in disguise.

• Finance — series and annuities. The present value of a stream of payments is $\sum_{i=1}^{n} \dfrac{C}{(1 + r)^i}$. Insurance premiums and pension calculations all run on this sum.

• Computer science — algorithm complexity. The running time of a nested loop is often $\sum_{i=1}^{n} i = \dfrac{n(n+1)}{2}$ — 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

Three errors account for most of the marks lost on sigma problems.

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: $\sum_{i=3}^{5} i = 1 + 2 + 3 + 4 + 5 = 15$.

The correct way: $\sum_{i=3}^{5} i = 3 + 4 + 5 = 12$. The lower bound is non-negotiable.

Mistake 2: Treating $\sum (a_i \cdot b_i)$ as $(\sum a_i) \cdot (\sum b_i)$.

Where it slips in: The sum rule splits sums of terms; it does not split products.

Don't do this: $\sum_{i=1}^{3} (i \cdot i) = \left(\sum_{i=1}^{3} i\right) \cdot \left(\sum_{i=1}^{3} i\right) = 6 \cdot 6 = 36$.

The correct way: $\sum_{i=1}^{3} i^2 = 1 + 4 + 9 = 14$. Sigma distributes over addition, not over multiplication.

Mistake 3: Forgetting that the dummy variable is local.

Where it slips in: Students try to "substitute" a value for $i$ outside the sum, treating $i$ like a free variable.

Don't do this: After writing $S = \sum_{i=1}^{n} i$, students set $i = n$ in the result.

The correct way: $i$ does not survive the sum. Once the sum is evaluated, the variable disappears entirely — the result depends only on $n$.

[IMAGE PROMPT 2 — A clean infographic split into three rows showing the same sum $\sum_{i=1}^{5} 2i$ written three ways: (1) as the expanded form $2 + 4 + 6 + 8 + 10$, (2) as sigma notation, (3) as the value $30$. Arrows connect the three forms.]

The Mathematician Who Shaped The Notation

Leonhard Euler (1707–1783, Switzerland). Euler standardised the use of $\Sigma$ for summation in his 1755 textbook Institutiones Calculi Differentialis. He also introduced the symbols $f(x)$ for a function, $e$ for the base of natural logarithms, and $i$ for the imaginary unit. The compact way mathematicians write sums today is almost entirely Euler's design.

Conclusion

• Sigma notation compresses a long sum into a compact expression with four pieces: $\Sigma$, the index variable, the bounds, and the summand.

• The four core formulas cover most exam problems: sums of $1$, $i$, $i^2$, $i^3$.

• Five properties make sigma rearrangeable like algebra.

• The most common slip is ignoring the lower bound — read it before you start computing.

• Sigma is the bridge from arithmetic to calculus, statistics, finance, and computer science.

A practical next step

Three problems to practise. If you stall on any of them, come back to the worked examples above.

1. Evaluate $\sum_{i=1}^{6} (3i - 1)$.

2. Evaluate $\sum_{k=1}^{15} k$.

3. Evaluate $\sum_{i=1}^{10} i^2$ using the closed-form formula, then verify by summing the first ten squares directly.

Want a Bhanzu trainer to walk through more sigma problems live? Book a free demo class — online globally, or in person at our McKinney, TX center.

For the arithmetic-sequence foundation that the first summation formula generalises, see Bhanzu's Arithmetic Sequence — Formula, Definition, Examples primer.