IB Maths AI HLSequences & SeriesPaper 1 & 2Σ notation~6 min read
Sigma Notation
Sigma notation is a compact way to write a sum. The symbol Σ means “add up”, a formula tells you each term, and two limits tell you where the sum starts and stops. It is the same idea as Sn, written far more efficiently.
📘 What you need to know
Σ is the Greek capital letter “sigma” and stands for sum.
The expression after Σ is the term formula, written in the index letter (usually r or k).
The lower limit is the first value of the index; the upper limit is the last.
To evaluate, substitute each index value from lower to upper, then add the terms.
The lower limit is not always 1 — it can be 0, 3, 7, or any integer.
Your GDC can compute sigma sums — a quick way to check your answer.
What is sigma notation?
Sigma notation packs a whole sum into one symbol. The term formula after Σ is the nth-term rule written in the index letter; the lower and upper limits say which terms to start and stop on.
The limits fix where the sum runs; the formula gives each term. Substituting r = 1 to 4 into 2r + 1 and adding gives 24.
Sigma notationΣr=1nur = u1 + u2 + … + unread from the lower limit up to the upper limit, summing the term formula
Evaluating a sigma expression
To evaluate, substitute each index value from the lower limit up to the upper limit into the term formula, then add the results. The number of terms is (upper limit − lower limit + 1).
Watch the lower limit — if it is 0, 3 or 7 rather than 1, your first term uses that value, not 1.
Writing a series in sigma notation
Going the other way, a written-out sum can be compressed into sigma notation. Find the term formula — the rule for the rth term — then set the limits to the first and last index values.
Finding the upper limit: set the term formula equal to the last term of the series and solve for the index — that value is the upper limit.
🧠Recipe — evaluating or writing sigma notation
Read the limits — the lower limit is the first index value, the upper the last.
Read the term formula — the expression after Σ, written in the index letter.
To evaluate, substitute each index value from lower to upper into the formula.
Add all those terms to get the total.
To write a series in sigma form, find the term formula, then set the limits to the first and last index values.
the index k runs from 3 to 6k=3: 9, k=4: 16k=5: 25, k=6: 36add the four terms9 + 16 + 25 + 36= 86start at k = 3, not 1 — the lower limit sets the first term.
WE 3
Writing a series in sigma notation
Write the series 4 + 7 + 10 + 13 + … + 31 using sigma notation.
find the term formula — step is 3rth term = 3r + 1 (check: r=1 ⇒ 4)find the upper limit — last term is 313r + 1 = 31 ⇒ r = 10Σr=110 (3r + 1)the formula gives each term; solving for the last term gives the upper limit.
WE 4
A partial sum of a sequence
A sequence is defined by un = 5 × 2n−1. Write u1 + u2 + … + u8 using sigma notation.
use the term formula, written in kuk = 5 × 2k−1the sum runs from k = 1 to k = 8Σk=18 (5 × 2k−1)replace n with the index letter; the subscripts of the first and last terms become the limits.
WE 5
A two-part term formula
Evaluate Σr=25 (r2 − r).
substitute r = 2, 3, 4, 5 into r2 − rr=2: 4−2 = 2, r=3: 9−3 = 6r=4: 16−4 = 12, r=5: 25−5 = 20add the four terms2 + 6 + 12 + 20= 40work out each term in full before adding — the whole formula is one term.
WE 6
Full question: write and evaluate
A sequence is defined by un = 3n + 4. (a) Write u1 + u2 + … + u6 in sigma notation. (b) Write u5 + u6 + … + u10 in sigma notation. (c) Evaluate the sum in part (a).
(a) term formula 3k + 4, limits 1 to 6Σk=16 (3k + 4)(b) same formula, limits 5 to 10Σk=510 (3k + 4)(c) terms for k = 1 to 67 + 10 + 13 + 16 + 19 + 22(a) & (b) above · (c) = 87only the limits change between (a) and (b) — the term formula stays the same.
💡 Top tips
Σ just means “add up” — it is shorthand for a sum, nothing more.
Always check the lower limit — it is often, but not always, 1.
The number of terms is upper − lower + 1 — useful as a quick check.
The index letter (r or k) is just a label — the answer does not depend on which is used.
Use your GDC’s sigma function to check sums quickly.
âš Common mistakes
Starting at r = 1 when the lower limit is something else.
Stopping one term early — the upper limit term is included.
Miscounting the terms — remember upper − lower + 1, not upper − lower.
Splitting a two-part formula — r2 − r is one term per value of r.
Wrong upper limit when writing a series — solve term formula = last term to find it.
Next up: Arithmetic Sequences & Series — sequences with a constant common difference, and the formulae for their terms and sums. Sigma notation often appears there as a tidy way to write the series before you sum it.
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