Arithmetic Sequences and Series

Arithmetic Sequences

Exercise:  Find the next term and the general formula for the following:

1. {2, 5, 8, 11, 14, ...}

2. {0, 4, 8, 12, 16, ...}

3. {2, -1, -4, -7, -10, ...}

For each of these three sequences there is a common difference.  In the first sequence the common difference is d = 3, in the second sequence the common difference is d = 4, and on the third sequence the common difference is d = -3.  We will call a sequence an arithmetic sequence if there is a common difference.

The general formula for an arithmetic sequence is

 an = a1  + (n - 1)d

Example

What is the difference between the fourth and the tenth terms of

{2,6,10,14,...)

We have

a10 - a4  =  (10 - 4)d  =  6(4)  =  24

Arithmetic Series

First we see that

1+ 2 + 3 + ... + 100  =  101 + 101 + ... + 101 (50 times)  =  101(50)

In general

n(n + 1)
1 + 2 + 3 + ... + n  =
2

Example

What is

S  =  1 + 4 + 7 + 10 + 13 +... +  46

Solution

S  =  1 + (1 + 1(3)) + (1 + 2(3))  + (1 + 3(3)) + ... + (1 + 15(3))

=  (1 + 1 + ... + 1) + 3(1 + 2 + 3 + ... + 15)

=  16 + 3(15)(16)/2

In General

d(n - 1)(n)
Sn = n (a1)+
2

= 1/2 [2n(a1) + d(n - 1)(n)]

= 1/2[2n(a1)+ dn2 - dn]

= (n/2)[2(a1)+ dn - d]= (n/2)[2(a1) + d(n - 1)]

Or Alternatively

 Sn = n/2(a1 + an)

Example

How much will I receive over my 35 year career if my starting salary is \$40,000, and I receive a 1,000 salary raise for each year I work here?

Solution

We have the series:

40,000 + 41,000 + 42,000 + ... + 74,000

=  35/2 (40,000 + 74,000)  =  \$1,995,500

For an interactive lesson on how to determine a term of an arithmetic sequence given two other terms Click Here