Basic Properties of Summation

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 can get out from the 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

[sum of (a_k +- b_k)] can be split into [sum of a_k] +- [sum of b_k].

The plus minus signs (±) are not affected
by the summation.

So [sum of (ak ± bk)] can be split into
[sum of ak] ± [sum of bk].

This formula is also needed
to solve the summation problems.

Proofs

Constant Multiple Rule in Summation: Proof of the Formula

Expand the summation.

Sigma notation

Factor the right side
by using the constant c.

Write a1 + a2 + ... + an
using the sigma notation.

Then, as you can see,
the constant c is out from the summation.

Sum Rule (Subtraction Rule) in Summation: Proof of the Formula

Expand the summation.

Sigma notation

Group sequence an and sequence bn separately.

Write each sequence using the sigma notation.

Then [sum of (ak ± bk)] is be split into
[sum of ak] ± [sum of bk].

Example

If (sum of a_k as k goes from 1 to n) = 7 and (sum of b_k as k goes from 1 to n) = -3, find the value of the given series. The sum of (5a_k + 2b_k) as k goes from 1 to n

As you've seen above,
the constants and the plus minus signs
are not affected by the summation.

So rewrite [sum of (5ak + 2bk)]
into 5⋅[sum of ak] + 3⋅[sum of bk].

[sum of ak] = 7
[sum of bk] = -3

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

Then (given) = 5⋅7 + 2⋅(-3).