IB Maths AI SL Topic 1 — Sequences & Series Paper 1 & 2 GDC-friendly ~5 min read

Sigma Notation

Sigma notation (the Σ symbol) is a compact way to write a sum. Three pieces tell you everything: where to start, where to stop, and what to add.

📘 What you need to know

Σ = capital Greek letter “sigma”, meaning “sum”.
Below Σ: the lower limit (where the index starts) — e.g. r = 1.
Above Σ: the upper limit (where the index stops).
After Σ: the expression — substitute the index and add up.
Any letter for the index — r, k, n, i — all mean the same thing.
Lower limit isn’t always 1 — check it carefully before you start.

Anatomy of sigma notation

Standard form ∑n = upperr = lower f(r) = f(lower) + f(lower+1) + … + f(upper)

Read the lower limit FIRST — that’s your starting value of r. Step it up one at a time until you hit the upper limit, substituting each into the expression.

Evaluating a sigma sum

Two methods:

By hand — list out each term and add. Works for any short sum.
On the GDC — most calculators have a Σ function. Type the expression, the variable, the lower limit, and the upper limit. Useful for long sums or to check by hand.

GDC tip: on TI-Nspire / Casio Classpad, the Σ template is under the maths/template menu. Always check the lower limit on the GDC matches the lower limit in the question.

🧭 Recipe — work with any sigma sum

Identify lower limit, upper limit, and expression.
Count the terms: number of terms = upper − lower + 1.
Substitute each value of the index into the expression.
Add all the resulting terms.
Use the GDC’s Σ to verify, especially for sums with 6+ terms.

Worked examples

WE 1

Evaluate a basic sigma sum

Evaluate ∑4r=1 (2r + 1).

Substitute r = 1, 2, 3, 4 into (2r + 1) r=1: 2(1)+1 = 3 r=2: 2(2)+1 = 5 r=3: 2(3)+1 = 7 r=4: 2(4)+1 = 9 Add 3 + 5 + 7 + 9 = 24 = 24

WE 2

Non-1 lower limit

Evaluate ∑6k=3 k².

Lower limit is 3, NOT 1 — start at k = 3 k=3: 3² = 9 k=4: 4² = 16 k=5: 5² = 25 k=6: 6² = 36 Add 9 + 16 + 25 + 36 = 86 = 86 always read the lower limit — starting at k=1 here would have given the wrong sum.

WE 3

Write a sum in sigma form

Write 5 + 10 + 15 + 20 + 25 + 30 using sigma notation.

Spot the pattern each term = 5 × (its position) expression: 5n Set the limits 6 terms → n goes from 1 to 6 Σ(n=1 to 6) 5n check: 5(1)+5(2)+…+5(6) = 5+10+15+20+25+30 ✓

WE 4

Convert u_n formula to sigma and evaluate

A sequence has u_n = 3n − 2. Write u₁ + u₂ + … + u₈ in sigma notation, then evaluate.

Write in sigma form Σ(n=1 to 8) (3n − 2) Substitute and add terms: 1, 4, 7, 10, 13, 16, 19, 22 sum = 1+4+7+10+13+16+19+22 Σ(n=1 to 8)(3n−2) = 92

WE 5

Long sum — use the GDC

Use your GDC to evaluate ∑20r=1 (r² + 1).

Enter on GDC’s Σ template expression: r² + 1 variable: r lower: 1, upper: 20 GDC returns = 2890 20 terms by hand would take ages — let the GDC do it.

WE 6

Real-world — savings using sigma

A savings plan adds $(3n − 1) to an account in week n. Use sigma notation to find the total saved over the first 10 weeks.

Set up sigma total = Σ(n=1 to 10) (3n − 1) Evaluate (GDC or by hand) terms: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29 sum = 155 total = $155 sigma is just shorthand — once you can set it up, the GDC does the adding.

💡 Top tips

Always read the lower limit — it’s not always 1.
Number of terms = upper − lower + 1 — use this to sanity-check.
The letter is arbitrary: r, k, n — all give the same sum if the limits and expression match.
Use the GDC’s Σ for any sum with more than ~6 terms — saves time and avoids arithmetic slips.

⚠ Common mistakes

Starting at the wrong index: ∑_k=3 means start at k = 3, not k = 1.
Miscounting terms: ∑ from 7 to 14 has 8 terms, not 7. (14 − 7 + 1 = 8.)
Forgetting brackets in the expression: ∑(2r + 1) is different from ∑2r + 1 — the first sums (2r+1), the second sums 2r then adds 1 once.
Substituting the limit value once instead of every index: you have to substitute EACH value of r from lower to upper.

Up next: Arithmetic Sequences & Series — when terms go up (or down) by a fixed amount each time. Comes with two ready-made formulae so you don’t have to add term-by-term.

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