# Basic Properties of Summation

How to use the basic properties of summation to solve summation problems: formulas, proofs, example, and its solution.

## Constant Multiple Rule in Summation

The constant *c* is not affected by the summation.

So *c* can get out from the summation.

This formula is needed

to solve the summation problems.

## Sum Rule (Subtraction Rule) in Summation

The plus minus signs (±) are not affected

by the summation.

So [sum of (*a*_{k} ± *b*_{k})] can be split into

[sum of *a*_{k}] ± [sum of *b*_{k}].

This formula is also needed

to solve the summation problems.

## Proofs

Expand the summation.

Sigma notation

Factor the right side

by using the constant *c*.

Write *a*^{1} + *a*^{2} + ... + *a*^{n}

using the sigma notation.

Then, as you can see,

the constant *c* is out from the summation.

Expand the summation.

Sigma notation

Group sequence *a*^{n} and sequence *b*^{n} separately.

Write each sequence using the sigma notation.

Then [sum of (*a*_{k} ± *b*_{k})] is be split into

[sum of *a*_{k}] ± [sum of *b*_{k}].

## Example

As you've seen above,

the constants and the plus minus signs

are not affected by the summation.

So rewrite [sum of (5*a*_{k} + 2*b*_{k})]

into 5⋅[sum of *a*_{k}] + 3⋅[sum of *b*_{k}].

[sum of *a*_{k}] = 7

[sum of *b*_{k}] = -3

Put these values into the series. (blue, brown)

Then (given) = 5⋅7 + 2⋅(-3).