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3.6PureCore

Sigma Notation

Sigma notation, ∑\sum, is a compact way to write a sum. Reading the limits tells you where to start and stop, and the series formulas let you evaluate the whole sum quickly.

25 min Sequences and Series
Edexcel A level Maths: 3.6 Sigma NotationCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Read and interpret sigma notation
  • Identify the first term, last term and number of terms
  • Evaluate arithmetic and geometric sums in sigma form
  • Split and shift summation limits
1

Reading the notation

In ∑r=1nf(r)\sum_{r=1}^{n} f(r), the variable rr runs from the bottom value up to the top value, and you add f(r)f(r) for each. The bottom and top are the limits\textbf{limits}.

∑r=14(2r+1)=3+5+7+9=24\sum_{r=1}^{4} (2r + 1) = 3 + 5 + 7 + 9 = 24
Substitute each rr and add.
2

Number of terms

The number of terms in ∑r=ab\sum_{r=a}^{b} is b−a+1b - a + 1 (not b−ab - a). Getting this right is essential before applying a series formula.

number of terms=(top)−(bottom)+1\text{number of terms} = (\text{top}) - (\text{bottom}) + 1
e.g. r=1r = 1 to 2020 is 20 terms.

Tip — ∑ from r = 5 to 12 has 12 − 5 + 1 = 8 terms — not 7.

3

Evaluating with formulas

Recognise whether the terms form an arithmetic or geometric series, find aa, dd or rr and nn, then apply the relevant sum formula. Sums that do not start at r=1r = 1 can be handled by subtracting: ∑r=ab=∑r=1b−∑r=1a−1\sum_{r=a}^{b} = \sum_{r=1}^{b} - \sum_{r=1}^{a-1}.

Example 1How many terms in ∑r=512arr\sum_{r=5}^{12} ar^{r}? (just the count)
112−5+112 - 5 + 1.
Answer
88 terms

Formula recap

∑r=abf(r)\sum_{r=a}^{b} f(r)
Sum of f(r) from a to b.
terms=b−a+1\text{terms} = b - a + 1
Counting terms.
∑r=ab=∑r=1b−∑r=1a−1\sum_{r=a}^{b} = \sum_{r=1}^{b} - \sum_{r=1}^{a-1}
Shifting the start.

Common mistakes to avoid

Counting b − a terms.
It is b − a + 1 (both endpoints are included).
Applying a series formula with the wrong n.
Count the terms correctly before substituting.

Key takeaways

  • ∑ from r = a to b adds f(r) for each r.
  • Number of terms = b − a + 1.
  • Use arithmetic/geometric sum formulas; subtract to handle a start above 1.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ∑r=14(2r+1)\sum_{r=1}^{4} (2r + 1) equals:

    • 2424
    • 2020
    • 1616
    • 1010
    Show answer and explanation

    Answer: 2424

    3 + 5 + 7 + 9.

  2. The number of terms in ∑r=512\sum_{r=5}^{12} is:

    • 88
    • 77
    • 1212
    • 1717
    Show answer and explanation

    Answer: 88

    12 − 5 + 1.

  3. In sigma notation, the top and bottom numbers are the:

    • Limits
    • Terms
    • Ratios
    • Differences
    Show answer and explanation

    Answer: Limits

    They bound the variable.

  4. ∑r=1n\sum_{r=1}^{n} has how many terms?

    • nn
    • n−1n - 1
    • n+1n + 1
    • 2n2n
    Show answer and explanation

    Answer: nn

    n − 1 + 1 = n.

  5. To evaluate a sigma sum you first identify whether it is:

    • Arithmetic or geometric
    • Linear or quadratic
    • Even or odd
    • Convergent only
    Show answer and explanation

    Answer: Arithmetic or geometric

    To pick the right formula.

  6. ∑r=ab\sum_{r=a}^{b} can be written as:

    • ∑1b−∑1a−1\sum_{1}^{b} - \sum_{1}^{a-1}
    • ∑1b+∑1a\sum_{1}^{b} + \sum_{1}^{a}
    • ∑1a−∑1b\sum_{1}^{a} - \sum_{1}^{b}
    • ∑1b−a\sum_{1}^{b-a}
    Show answer and explanation

    Answer: ∑1b−∑1a−1\sum_{1}^{b} - \sum_{1}^{a-1}

    Subtract the part before a.

Next step

Check your understanding

Work through questions on Sigma Notation. Use the feedback to decide what needs another look.

1 past-paper question · 5 marks

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