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Arithmetic Sequences
This section will define and discuss arithmetic sequences, nth term formulas, and the sum formulas for finite arithmetic series.
An arithmetic sequence may be defined as:
A sequence { an } is arithmetic if each pair of consecutive terms differs by the same amount, d = a i –^ a i – 1. The number d is called the common difference in the sequence.
(Note: When the formula given in this definition is rewritten as a i = a i – 1 + d it is known as a recursive formula because it defines a given term by referencing a proceeding term.)
The following example will show how to find the common difference of an arithmetic sequence.
Example 1: Find the common difference of the arithmetic sequence an = 3 – 5 n.
Solution: Step 1: Substitute. Substitute the example formula into the definition formula.
1 3 5 3 5 1
d a i ai i i
= ⎡⎣^ − ⎤⎦^ − ⎡⎣ − − ⎤⎦
Step 2: Solve for d.
d ( i ) ( i )
i i
The common difference for this arithmetic sequence is d = –5.
The nth term of an arithmetic sequence is defined as:
The nth term of an arithmetic sequence, whose first term is a 1 and whose common difference is d , is given by the formula an = a 1 + ( n – 1) d.
The following examples will show how to find specific terms of an arithmetic sequence.
Example 2: Find the first four terms and then the twentieth term of the arithmetic sequence whose first term is –1 and whose common difference is 4. Solution: Step 1: Substitute. Since it was given that a 1 = –1 and d = 4, the solutions for the requested terms are found by substitution into the defined formula:
[ ]
a n a n d a GIVEN a a a a
1 1 2 3 4 20
Step 2: Solve.
a a a a a
1 2 3 4 20
Example 3: Find the thirty-eighth term of the arithmetic sequence whose first term is 8 and whose nth term is given by a i+1 = ai – 7. Solution: Step 1: Solve for d.
( )
i i i i
d a a a a
1 7 7
Example 4 (Continued):
Step 3: Solve for a 18.
The values found for d and a 1 are used to solve for a 18.
a a n d a a a a
18 1 18 18 18 18
There are formulas for determining the sum of an arithmetic sequence.
( )
1
1
n
n n
. S n^ a n 1 . S n a a
d
The next two examples will show how to use these formulas.
Example 5: Find the sum of the first 99 terms of the arithmetic sequence whose nth term is
n a 7 n = + (^) 2.
Solution:
Step 1: Analysis.
The terms given by the problem or that are obvious are:
n = 99 , a (^) 1 = 7 +^12 , and a 99 = 7 +^929
Example 5 (Continued):
Step 2: Substitute and solve.
Sn n ( a an )
S
1
99
= ⎡^ ⎛^ + ⎞^ + ⎛ + ⎤
= ⎛⎜^ + ⎞⎟
= ⎛⎜^ ⎞⎟
Example 6: A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $7500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation.
Solution:
Step 1: Analysis.
The terms that are given by the problem or that are obvious are:
n = 10 ;a 1 = 10 000 , ; and d = 7500
Step 2: Substitute and solve.
[ ]
n S n a n d
S , , , $ ,
1
10
= ⎡⎣^ + − )⎤⎦
