ARITHMETIC SEQUENCES

A sequence is a list of numbers that may form a pattern. Each number in a sequence is a term.

When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. The distances in the table form an arithmetic sequence with d = 0.2.

The variable a is often used to represent terms in a sequence. The variable a₇, read “a sub 7,” is the seventh term in a sequence. To designate any term, or the n^th term, in a sequence, you write an, where n can be any number.

To find a term in an arithmetic sequence, add d to the previous term.

Finding a Term of an Arithmetic Sequence

The n^th term of an arithmetic sequence with common difference d is

a_n  =  a_n-1 + d

Identifying Arithmetic Sequences

Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence.

Example 1 : 

15, 10, 5, 0, ........

Solution : 

Step 1 :

15, 10, 5, 0, ........

Find the difference between successive terms.

10 - 15  =  -5

5 - 10  =  -5

0 - 5  =  -5

The common difference is -5.

Step 2 :

Use the common difference to find the next 3 terms.

Add -5 to each term to find the next term.

a_n  =  a_n-1 + d

0 + (-5)  =  0 - 5  =  -5

-5 + (-5)  =  -5 - 5  =  -10

-10 + (-5)  =  -10 - 5  =  -15

The sequence appears to be an arithmetic sequence with a common difference of -5.

The next 3 terms are -5, -10, -15.

Example 2 : 

2, 5, 10, 17, ........

Solution : 

2, 5, 10, 17, ........

Find the difference between successive terms.

5 - 2  =  3

10 - 5  =  5

17 - 10  =  7

The difference between successive terms is not the same.

This sequence is not an arithmetic sequence.

Finding the nth Term of an Arithmetic Sequence

Find the indicated term of each arithmetic sequence.

Example 3 :

25^th term : 6, 2, -2, -6,.........

Solution : 

Step 1 : Find the common difference.

2 - 6  =  -4

-2 - 2  =  -4

-6 - (-2)  =  -4

The common difference is -4.

Step 2 : Find the 25^th term.

Write the rule to find the n^th term.  

a_n  =  a₁ + (n - 1)d

Substitute 6 for a₁, 25 for n, and -4 for d.

a₂₅  =  6 + (25 - 1)(-4)

Simplify the expression in parentheses.

a₂₅  =  6 + (24)(-4)

Multiply. 

a₂₅  =  6 - 96

Subtract. 

a₂₅  =  -90

Example 4 :

16^th term : a₁ = 8; d = 5

Solution : 

Write the rule to find the n^th term.  

a_n  =  a₁ + (n - 1)d

Substitute 8 for a₁, 16 for n, and 5 for d.

a₁₆  =  8 + (16 - 1)(5)

Simplify the expression in parentheses.

a₁₆  =  8 + (15)(5)

Multiply. 

a₁₆  =  8 + 75

Add. 

a₁₆  =  83

Travel Application

Example 5 :

The odometer on a car reads 60,180 on day 1. Every day, the car is driven 60 miles. If this pattern continues, what is the odometer reading on day 22?

Solution : 

Notice that the sequence for the situation is arithmetic with d = 60 because the odometer reading will increase by 60 miles per day. Since the odometer reading on day 1 is 60,180 miles, a₁ = 60,180. Because you want to find the odometer reading on day 22, you will need to find the 20^th term of the sequence, so n = 22.

Write the rule to find the n^th term.  

a_n  =  a₁ + (n - 1)d

Substitute 60,180 for a₁, 22 for n, and 60 for d.

a₂₂  =  60,180 + (22 - 1)(60)

Simplify the expression in parentheses.

a₂₂  =  60,180 + (21)(60)

Multiply. 

a₂₂  =  60,180 + 1,260

Add. 

a₂₂  =  61,440

The odometer will read 61,440 miles on day 22.

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