Sigma Notation (Edexcel IGCSE Further Pure Maths)

Revision Note

Written by: Roger B

Reviewed by: Dan Finlay

Sigma notation is used to show the sum of a certain number of terms in a sequence
- The symbol Σ is the capital Greek letter sigma
Σ stands for ‘sum’
- The expression to the right of the Σ tells you what is being summed
- The limits above and below tell you which terms you are summing
- $r$ and $k$ are both common variables to use when writing sigma sums

Be careful, the limits don’t have to start with 1
- For example $\sum_{k = 0}^{4} (2 k + 1)$ or $\sum_{k = 7}^{14} (2 k - 13)$
You need to be able to read and use sigma notation when answering questions about series
- For example the sum of the fifth through ninth terms of an arithmetic series
  - This could be written as $\sum_{k = 5}^{9} (a + (k - 1) d)$
- Or the sum of the first $n$ terms of a geometric series
  - This could be written as $\sum_{k = 1}^{n} a r^{k - 1}$

Your calculator may be able to use sigma notation
- If so make sure you know how it works
- You can use this to check your work

The terms of a series are defined by $u_{n} = 2 \times 3^{n - 1}$ , for $n = 1, 2, 3, . . .$ .

(a) Write an expression for the sum of the first six terms of the series using sigma notation.

Use sum limits from 1 to 6

$\sum_{k = 1}^{6} u_{k}$

Note that ' $n$ ' is replaced by ' $k$ ' to match the variable used for the sigma sum

Now replace $u_{k}$ with the actual formula, being sure to use $k$ instead of $n$ again

$\sum_{k = 1}^{6} (2 \times 3^{k - 1})$

(b) Write an expression for the sum of the seventh through twelfth terms of the series using sigma notation.

This will be the same as in part (a), except that the limits will be 7 on the bottom and 12 on the top

$\sum_{k = 7}^{12} (2 \times 3^{k - 1})$

This is very similar to the above, but the sum limits will start at 1 and go to $n$

$\sum_{k = 1}^{n} u_{k}$

Be careful here – $k$ is the variable for the sigma sum, and $n$ means we're going up to the $n$ ^th term

$\sum_{k = 1}^{n} (2 \times 3^{k - 1})$

Last updated: 16 October 2024

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