12.2 Analyze Arithmetic Sequences and Series

 

Arithmetic sequence

A sequence in which the difference between consecutive terms is constant

 

Common difference

The constant difference between terms of an arithmetic sequence, denoted by d

 

Arithmetic series

The expression formed by adding the terms of an arithmetic sequence, denoted by Sn

 

Example 1

Identify arithmetic sequences

 

Tell whether the sequence -5, -3, -1, 1, 3,... is arithmetic.

Find the differences of consecutive terms.

a2 - a1 = _-3 - (-5)_ = _2_

a3 - a2 = _-1 - (-3)_ = _2_

a4 - a3 = _1- (-1)_ = _2_

a5 - a4 = _3 - 1_ = _2_

Each difference is _2_, so the sequence _is_ arithmetic.

 

Decide whether the sequence is arithmetic.

 

1.      32, 27, 21, 17, 10, . . .

not arithmetic

 

RULE FOR AN ARITHMETIC SEQUENCE

The nth term of an arithmetic sequence with first term a1 and common difference d is given by:

an = a1 + (n - 1)d

 

 

Example 2

Write a rule for the nth term

 

Write a rule for the nth term of the sequence. Then find a19.

a. 2, 9, 16, 23, . . .                      b. 57, 45, 33, 21, . . .

 

Solution

a.   The sequence is arithmetic with first term a1 = 2 and common difference
d = _9 - 2_ = _7_. So, a rule for the nth term is:

an = a1 + (n - 1)d                   Write general rule.

= _2_ + (n - 1) _7_           Substitute for a1 and d.

= _-5 + 7n_                       Simplify.

The 19th term is a19 = _-5 + 7(19)_ = _128_.

 

b.   The sequence is arithmetic with first term a1 = 57 and common difference
d = _45 - 57_ = _-12_. So, a rule for the nth term is:

an = a1 + (n - 1)d                   Write general rule.

= _57_ + (n - 1) (_-12_)   Substitute for a1 and d.

= _69 - 12n_                     Simplify.

The 19th term is a19 = _69 - 12(19)_ = _-159_.

 

Write a rule for the nth term of the arithmetic sequence. Then find a22.

2.     9, 5, 1, -3, . . .

an = 13 - 4n, - 75

 

3.     -15, -9, -3, 3, . . .

an = -21 + 6n, 111

 

Example 3

Write a rule given a term and common difference

 

One term of an arithmetic sequence is a11 = 41. The common difference is d = 5.

(a) Write a rule for the nth term, (b) Graph the sequence.

a.    Use the general rule to find the first term.

an = a1 + (n - 1)d                           Write general rule.

_41_ = a1 + (_11_ - 1) _5_                Substitute for an, n, and d.

_-9_ = a1                                                       Solve for a1.

So, a rule for the nth term is:

an = _-9_ + (n - 1) _5_                      Substitute for a1 and for d.

= _-14 + 5n_                                  Simplify.

 

b.   Create a table of values for the sequence. Notice that the points lie on a line.

 


n

1

2

3

4

5

6

an

_-9_

_-4_

_1_

_6_

_11_

_16_

 

Example 4

Write a rule given two terms

 

Two terms of the arithmetic sequence are a6 = 7 and a22 = 87. Find a rule for the nth term.

1.   Write a system of equations using an = a1 + (n - l)d and substituting 22 for n (Equation 1) and then 6 for n (Equation 2).

a22 = a1 + (22 - l)d                       _87_ = a1 + _21_ d

a6 = a1 + (6 - l)d                         _7__ = a1 + _5_ d

2.    Solve the system.                   _80_ =        _16_ d

_5_ = d

_87_ = a1 + _21_ (_5_)

_-18_ = a1

 

3.    Find a rule for an.             an = a1 + (n - 1) d

= _-18_ + (n - 1) _5_

= _-23 + 5n_

 

 

THE SUM OF A FINITE ARITHMETIC SERIES

The sum of the first n terms of an arithmetic series is:

 


In words, Sn is the _mean_ of the _first and nth_ terms, _multiplied_ by
_the number of terms_.

 

Example 5

Find a sum

 


Find the sum of the arithmetic series

 

a1 = 9 + 3(_1_) = _12_           Identify first term.

a15 = 9 + 3(_15_) = _54 _       Identify last term.

S15 =                                        Write rule for S15.

= _495_                             Simplify.