IB Maths AA HL Topic 1 — Number & Algebra Paper 1 & 2 ~7 min read

Sigma Notation

Sigma notation is just a compact way of writing a sum. Instead of writing out every term with plus signs, you tell the reader: “here’s the formula for a typical term, here’s where to start, and here’s where to stop”. Once you can read it fluently, every later note in this section (arithmetic, geometric, binomial theorem, integration sums) becomes much faster to work through — IB papers use Σ everywhere.

📘 What you need to know

The Greek letter Σ (capital sigma) means “sum”. Whatever follows the sigma is the formula for one term; the limits tell you where to start and stop.
The variable underneath (often r, k, or i) is the dummy variable — it counts up by 1 each step.
Lower limit tells you the starting value of the dummy variable. Upper limit tells you the final value (inclusive).
Lower limits don’t always start at 1 — read carefully.
Two HL-useful properties: constants come out (Σ cu_r = c·Σ u_r) and sums split (Σ(u_r + v_r) = Σ u_r + Σ v_r).
Your GDC computes Σ directly — useful for checking by-hand answers.

The anatomy of a sigma expression

Every Σ has three pieces: where it starts, where it stops, and what each term looks like. Once you can label these three parts, reading any sigma expression is mechanical.

The three parts of a sigma expression

The dummy variable’s letter doesn’t matter. r, k, and i are all common. As long as the same letter is used in the formula and the lower limit, the value of the sum is the same.

Computing a sum from sigma notation

Substitute every value of the dummy variable from the lower limit up to the upper limit into the formula, and add the results.

🧭 Recipe — evaluate a sigma expression

Read off the limits. Note the start value, the stop value, and the formula.
Substitute each value of the dummy variable into the formula one at a time.
Add all the results together.
Check with your GDC — most calculators have a Σ button that does the whole thing in one keystroke.

Example: Σ_r=1⁴ (2r + 1) = (2·1 + 1) + (2·2 + 1) + (2·3 + 1) + (2·4 + 1) = 3 + 5 + 7 + 9 = 24.

Writing a sum in sigma notation

Going the other way is also a common exam task. You’re given a list of terms, and you have to compress it into a sigma expression. The key step is finding the formula for a typical term in terms of the dummy variable.

🧭 Recipe — convert a sum into Σ notation

Spot the pattern — how does each term depend on its position?
Write a formula in terms of a dummy variable (call it k) for the general term.
Set the limits — what value of k gives the first term, and what gives the last?
Verify by substituting the lower and upper limits back in — they should produce the original first and last terms.

Example: the sum 6 + 11 + 16 + 21 + 26 has each term equal to 5k + 1, with k running from 1 to 5. So it equals Σ_k=1⁵ (5k + 1).

Watch the lower limit — it’s not always 1

A common trap: assuming Σ always starts at k = 1. It often does, but plenty of exam questions deliberately use other starting points (like k = 0, k = 3, or k = 7).

Same formula, different limits:
Σ_k=1⁴ k² = 1 + 4 + 9 + 16 = 30 whereas Σ_k=2⁵ k² = 4 + 9 + 16 + 25 = 54.

The number of terms in Σ_k=a^b is (b − a + 1), not just (b − a). Lots of students miscount by 1 — easy mistake to make under time pressure.

Two HL-useful properties

Two simple algebraic rules that often turn an awkward sum into something clean.

Constant out front Σ_r=1ⁿ (c · u_r) = c · Σ_r=1ⁿ u_r

Sum splits Σ_r=1ⁿ (u_r + v_r) = Σ_r=1ⁿ u_r + Σ_r=1ⁿ v_r

🤔 Why do these work?

They’re both consequences of how addition behaves. A constant inside every term factors out the same way c(a + b + c) = ca + cb + cc works in reverse. And rearranging the order of addition lets you group like terms — that’s the sum-split.

These two properties become essential when you reach the binomial theorem and integration. They turn a clumsy single sigma into two cleaner ones.

Using your GDC

Most modern graphing calculators have a sum function that takes the formula, dummy variable, lower limit, and upper limit, and returns the answer in one keystroke.

On a TI-Nspire: use the Σ() template under the menu > calculus options. On Casio fx-CG50: use SHIFT > (CALC) > Σ. Always double-check the limits are entered correctly.

Worked examples

WE 1

Compute a sum (lower limit 1)

Find the value of Σ_r=1⁵ (3r − 2).

Step 1: Substitute r = 1, 2, 3, 4, 5 r = 1: 3(1) − 2 = 1 r = 2: 3(2) − 2 = 4 r = 3: 3(3) − 2 = 7 r = 4: 3(4) − 2 = 10 r = 5: 3(5) − 2 = 13 Step 2: Add 1 + 4 + 7 + 10 + 13 = 35 = 35

WE 2

Lower limit not equal to 1

Find the value of Σ_k=2⁵ k².

Step 1: Substitute k = 2, 3, 4, 5 k = 2: 4 k = 3: 9 k = 4: 16 k = 5: 25 Step 2: Add 4 + 9 + 16 + 25 = 54 = 54 there are 4 terms here (5 − 2 + 1), not 3 — count carefully when limits don’t start at 1

WE 3

Write a sum using sigma notation

Express the sum 5 + 9 + 13 + 17 + 21 + 25 in sigma notation.

Step 1: Spot the pattern terms increase by 4 each time, starting at 5 general term: 4k + 1 Step 2: Find the limits k = 1: 4 + 1 = 5 ✓ (first term) k = 6: 24 + 1 = 25 ✓ (last term) = Σ_k=1⁶ (4k + 1)

WE 4

Sigma notation from a defined sequence

A sequence is defined by u_n = 4n + 3 for n ∈ ℤ⁺. Express u₅ + u₆ + u₇ + … + u₁₂ in sigma notation.

Step 1: Identify the formula and the limits general term: 4k + 3 starts at k = 5, ends at k = 12 = Σ_k=5¹² (4k + 3) note the lower limit is 5, not 1 — match the subscript of the first term you’re summing

WE 5

Apply the constant-out and split properties

Given that Σ_r=1¹⁰ r = 55 and Σ_r=1¹⁰ r² = 385, find Σ_r=1¹⁰ (2r² − 3r).

Step 1: Split the sum = Σ 2r² − Σ 3r Step 2: Pull constants out = 2 · Σ r² − 3 · Σ r Step 3: Substitute the given values = 2(385) − 3(55) = 770 − 165 = 605 the two properties together turn one messy sum into two clean ones

WE 6

How many terms does this sum contain?

How many terms are there in the sum Σ_k=4²⁰ (2k − 1)?

Apply the count formula number of terms = upper − lower + 1 = 20 − 4 + 1 = 17 17 terms the +1 catches the off-by-one error students often make

💡 Top tips

Always count terms as (upper − lower + 1). Forgetting the +1 is the most common slip.
Verify by substituting both limits when you build a sigma expression yourself — the lower limit should produce the first term, the upper limit the last.
If the sum has more than 5 or 6 terms, use your GDC to evaluate it. Faster and more reliable.
For complex sums, look for patterns to split. A sum like Σ(3r + 2r²) becomes 3·Σr + 2·Σr² — much easier.
The dummy variable letter doesn’t matter. You can replace k with r or i without changing the value.
If a sigma expression looks unfamiliar, write out the first 3 or 4 terms by hand — usually clarifies what’s going on.
For sigma sums of arithmetic or geometric sequences, the standard sum formulas from the next two notes are far faster than expanding term-by-term.

⚠ Common mistakes

Miscounting terms. Σ_k=3¹⁰ has 8 terms (10 − 3 + 1), not 7.
Assuming the lower limit is 1. Always read the bottom of the sigma carefully.
Using the wrong dummy variable. If the limit says k = 1 but the formula uses r, the sum is meaningless. Both must match.
Splitting a product like a sum. Σ(u_r · v_r) ≠ Σu_r · Σv_r. Only sums and constants behave nicely.
Forgetting that the upper limit is inclusive. Σ_k=1⁵ includes k = 5 — both endpoints count.
Pulling out a non-constant. Σ r · u_r ≠ r · Σ u_r. Only true constants come out — variables that depend on the index don’t.
GDC input errors. When typing into the calculator, use brackets liberally — particularly around the formula. Σ(3k+1, k, 1, 5) is the safe form.

Sigma notation is the IB’s universal shorthand for “sum these up”. The next two notes (arithmetic and geometric series) give you the closed-form formulas that turn a long sigma calculation into one line. Get fluent with the notation here, and the rest of the section reads much faster.

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