# Sigma Notation

How to solve sigma notation problems: meaning, examples, and their solutions.

## Meaning

Sigma notation is a simple way

to write the sum of a sequence (= series).

'Σ' is the greek capital letter 'sigma'.

The given summation means (and is read as)

'the sum of *a*_{i} as i goes from 1 to *n*'.

## Example 1

Find *a*_{k}.

1/[1⋅(1 + 1)]

1/[2⋅(2 + 1)]

1/[3⋅(3 + 1)]

...

1/[99⋅(99 + 1)]

So *a*_{k} = 1/[*k*(*k* + 1)].

1/[1⋅(1 + 1)]

1/[2⋅(2 + 1)]

1/[3⋅(3 + 1)]

...

1/[99⋅(99 + 1)]

So *k* goes from 1 to 99.

So the given series is

the sum of 1/[*k*(*k* + 1)]

as *k* goes from 1 to 99.

## Example 2

Find *a*_{i}.

4^{3}, 5^{3}, 6^{3}, ... 19^{3}

So *a*_{i} = *i*^{3}.

4^{3}, 5^{3}, 6^{3}, ... 19^{3}

So *i* goes from 4 to 19.

So the given series is

the sum of *i*^{3}

as *i* goes from 4 to 19.

## Example 3

Find *a*_{n}.

The given series is an arithmetic series

whose *a*_{1} is 1 and *d* is +2.

So *a*_{n} = 1 + (*n* - 1)⋅2

= 2*n* - 1.

Arithmetic sequences

There are 7 terms.

So *n* is from 1 to 7.

So the given series is

the sum of '2*n* - 1'

as *n* goes from 1 to 7.

## Example 4

*a*_{n} = 2*n* - 1

So the sigma notation shows the arithmetic series.

To use the arithmetic series formula,

find *a*_{1}, *d*, and *n*.*a*_{1} = 1*d* = 2 (coefficient of *n*)*n* = 10 (*i* is from 1 to 10.)

Arithmetic sequences

Use the arithmetic series formula.

(given) = *S*_{10}

= (10/2)[2⋅1 + (10 - 1)⋅2]

## Example 5

Find the pattern of the terms.*i*^{1} = *i**i*^{2} = -1*i*^{3} = -*i**i*^{4} = +1*i*^{5} = *i*^{4⋅1 + 1} = *i**i*^{6} = *i*^{4⋅1 + 2} = -1*i*^{7} = *i*^{4⋅1 + 3} = -*i**i*^{8} = *i*^{4⋅2 + 0} = +1

'*i*, -1, -*i*, 1' is repeating.*i*^{100} = *i*^{4⋅25 + 0} = 1

So '*i*, -1, -*i*, 1' is repeating 25 times.

Powers of *i*

*i* - 1 - *i* + 1 = 0

So each 'group' is 0.

So the sum of the 'groups' is 0.