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 Video by Zeeshan Zamurred Sequences and Series

Edexcel A level Maths: 3.6 Sigma NotationWatch the full walkthrough 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

Reading the notation

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

r = 1 \sum 4 (2 r + 1) = 3 + 5 + 7 + 9 = 24

Substitute each

r

and add.

Number of terms

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

number of terms = (top) - (bottom) + 1

e.g.

r = 1

20

is 20 terms.

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

Evaluating with formulas

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

12 - 5 + 1

Answer

8

terms

Formula recap

r = a \sum b f (r)

Sum of f(r) from a to b.

terms = b - a + 1

Counting terms.

r = a \sum b = r = 1 \sum b - r = 1 \sum 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.

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