Arithmetic Series

A series is the expression for the sum of the terms of a sequence.

Sequence Series6, 9, 12, 15, 18

6+9+12+15+18

3, 7, 11, 15, ... 3+7+11+15+...

Evaluate the seriesEx. 2, 11, 20, 29, 38, 472+11+20+29+38+47 = 147Ex. 100, 125, 150, 175, 200, 225100+125+150+175+200+225 = 975

An arithmetic series is a series whose terms form an arithmetic sequence. We have a formula for evaluating an arithmetic series easily.

a1+a2+a3+...+an = (a1+an)n2

Hint: a1 is the first term in the seriesan is the last term in the seriesn is the number of terms in the series

Evaluate the series using the formulaEx. 2, 11, 20, 29, 38, 47( )(2+47) = 147Ex. 100, 125, 150, 175, 200, 225 ( )(100+225) = 97562

62

Another way to write a series is called summation notation. Ʃ is the Greek letter sigma.

Ʃ

(5n+1)

n=1

n=3

explicit formula for the sequence

greatest value of n, the number of terms

least value of n,usually n=1

Write the explicit formula for the series 3+6+9+...for 33 terms.

sequence term n=#of term What do you have to do to n to get the sequence term?

3 1 n=1 3*1=3 so, 3n

6 2 n=2 3*2=6 so, 3n

9 3 n=3 3*3=9, so, 3n

Write the summation notation to write the series 3+6+9+...for 33 terms.

Ʃ

3n

n=1

n=33

Use the series Ʃ(5n+1).a. Find the number of terms in the series.

b. Find the first and last terms of the series.

c. Evaluate the series.

n=1

3

Since the values of n go from 1 to 3. There are three terms because n=1, n=2, and n=3.

The first term is n=1, so 5(1)+1=6. The last term is n=3, so 5(3)+1=16

1st term = 62nd term = 113rd term = 16

6+11+16 = 33

Using the calculator to evaluate the series.

Steps:1. Select MATH2. Select 0:summation Ʃ(3. Enter the information in4. Hint ENTER