The big idea: The symbol Σ (Greek capital sigma) means add up — read it from the bottom number up to the top, putting each value into the expression.
How to read it
Read each part of the sigma separately.
Bottom
Where the index starts (the lower limit).
Start at r = 1.
Top
Where the index stops (the upper limit).
Stop at r = 4.
Expression
What you add each time (the summand).
Add 2r + 1 each time.
How to evaluate a sum: Write the first and last terms, notice the gap is constant (so it is arithmetic), then use the arithmetic sum formula.
That is much faster than adding every term.
Count the terms first: The number of terms is (upper limit − lower limit + 1).
If r runs from 1 to 10, that is 10 − 1 + 1 = 10 terms.
IB-style question — evaluate a sigma sum
Evaluate ∑<sub>r=1</sub><sup>10</sup> (3r + 2).
Step by step
- Find the first and last terms by substituting r = 1 and r = 10.
- The terms go up by 3 each time, so it is arithmetic with n = 10.
- Use the sum formula with the first and last term.
Final answer
∑<sub>r=1</sub><sup>10</sup> (3r + 2) = 185.
Spotting it is arithmetic: If the summand is linear in the index (like 3r + 2), the terms form an arithmetic sequence.
The common difference is the coefficient of the index (here d = 3).
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A sum of a linear term is an arithmetic series: To evaluate Σ of a linear expression: first term = value at the lower limit; common difference = coefficient of the index; number of terms = upper − lower + 1. Then apply Sₙ.
IB-style question — limits that do not start at 1
Evaluate ∑<sub>r=3</sub><sup>12</sup> (4r − 1).
Step by step
- First and last terms: substitute r = 3 and r = 12.
- Number of terms.
- Apply the sum formula.
Final answer
∑<sub>r=3</sub><sup>12</sup> (4r − 1) = 290.
IB-style question — exact value with a logarithm
Evaluate this sum, giving an exact answer.
Step by step
- It is linear in r, so arithmetic. Read off the first term (r = 1) and the common difference (the coefficient of r).
- Last term (r = 10) and the number of terms.
- Use Sₙ = (n/2)(u₁ + uₙ) — keep ln 2 exact, never a decimal.
- Tidy up.
Final answer
90 − 55 ln 2 — leave it exact (logs and surds stay as they are).
IB-style question — another with a logarithm
Evaluate this sum, giving an exact answer.
Step by step
- Linear in r → arithmetic. First term (r = 1) and common difference (the coefficient of r).
- Last term (r = 5) and the number of terms.
- Use Sₙ = (n/2)(u₁ + uₙ); keep ln 2 exact.
- Tidy up — factor the 6 first if it helps.
Final answer
15 + 15 ln 2 (exact).
Common mistakes
- Counting terms as (upper − lower) and forgetting the +1.
- Assuming the index always starts at 1.
- Multiplying the summand instead of adding its values.
Do this instead
- Number of terms = upper − lower + 1.
- Substitute the actual lower limit for the first term.
- Add the substituted values — Σ means a sum.
The GDC shortcut (Paper 2): On Paper 2 you can evaluate any sigma sum directly with the calculator's sum(seq(…)) command — no formula needed.
On Paper 1 (no calculator) you must use the arithmetic sum formula by hand.