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12.2 Arithmetic Sequences and Series

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12.2 Arithmetic Sequences and Series

©2001 by R. Villar

All Rights Reserved

Arithmetic Sequences and Series

Arithmetic Sequence: sequence whose consecutive terms have a common difference.

Example:3, 5, 7, 9, 11, 13, ...

The terms have a common difference of 2.

The common difference is the number d.

Example: Is the sequence arithmetic? –45, –30, –15, 0, 15, 30

Yes, the common difference is 15

How do you find any term in this sequence?

To find any term in an arithmetic sequence, use the formulaan = a1 + (n – 1)dwhere d is the common difference.

Example: Find a formula for the nth term of the arithmetic sequence in which the common difference is 5 and the first term is 3.

an = a1 + (n – 1)d

a1 = 3 d = 5

an = 3 + (n – 1)5

Example: If the common difference is 4 and the fifth term is 15, what is the 10th term of an arithmetic sequence?

an = a1 + (n – 1)d

We need to determine what the first term is...

d = 4 and a5 = 15

a5 = a1 + (5 – 1)4 = 15

a1 = –1

a10= –1 + (10 – 1)4

a10 = 35

To find the sum of an arithmetic series, we can use summation notation.

Which can be simplified to:

Example: Find the sum of the first 100 terms of the arithmetic sequence 1, 2, 3, 4, 5, 6, ...

n = 100

= 5050