Using Sigma Notation

The sum of the first sixteen positive integers can be written

1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16,

or more briefly 1+2+…+16. It should be clear that each number in the sum is obtained by adding 1 to the preceding number. a more precise notation is

`sum_(i=1)^16`i.

In the same way.

`sum_(i=1)^6`f(i)=f(1)+f(2)+f(3)+f(4)+f(5)+f(6);

The notation means “first evaluate the expression after the `sum` (that is, f(i)) when i=1, then when i=2,…, then when i=6, and add the results.”

Concept - using sigma notation:

A shorthand notation for finding addition is to use the Greek capital sigma help `sigma`. This is particularly useful in statistics where there are many data values to be added mutually.

`sum_(n=5)^(15) n^3`

Where

15             At the top of the sigma, sign is the last value the variable can take

n=5           At the bottom of the sigma sign is the variable which servers as a counter and the first value it takes.

Shorthand for `5^3+6^3+* * *+(15)^3`.

Using sigma notation in arithmetic series with example problem:

Assume the `n^(th)` term of a series is `u_n`. Then the addition of the first 10 terms could be expressed as                  `

sum_(n=1)^10 u^n`

It is possible to say the addition of an arithmetic series in sigma notation help.

For instance, consider the arithmetic series

2+4+16+256+…

The nth term of this series is 2+`(n)^2`, or in a simpler form n2.

Hence the sum of the first 10 terms of the given series can be written as `sum_(n=2)^10 n2`

Example 1:

`sum_(x=3)^(5)x^2`

Solution:

The expression (using sigma notation help) to be summed here is x2.

You need to substitute x=3 then x=3… up to x=5

Addition the terms jointly.

`sum_(x=3)^(4)x^2`= 32+42 +52

=9+16+25

=25+25

=50

Example 2:

`sum_(K=2)^(5)K^2`

Solution:

The expression (using sigma notation help) to be summed here is K2.

You need to substitute K=2 then K=2… up to K=5

Adding the terms together.

`sum_(K=2)^(4)K^2`=22+32+42 +52

=4+9+16+25

=13+41

=54

Example 3:

`sum_(R=1)^(3)R^2`

Solution:

The expression (using sigma notation help) to be summed here is R2.

You need to substitute R=2 then R=1… up to R=3

Adding the terms together.

`sum_(R=1)^(3)R^2`=12+22+32

=1+4+9

=5+9

=14