Arithmetic Sequences and Series

Arithmetic Sequences and Series

Arithmetic Sequences

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

{2, 5, 8, 11, 14, ...}
{0, 4, 8, 12, 16, ...}
{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

a_n = 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)
        S_n = 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

S_n = n/2(a₁ + a_n)

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

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