Analyzing Arithmetic Sequences and Series

Section 8.2 beginning on page 417

Identifying Arithmetic SequencesIn an arithmetic sequence, the difference of consecutive terms is constant. This constant difference Is called common difference and is denoted by d.

Example 1: Tell whether each sequence is arithmetic.

a)

b)

7 7 7 7

Arithmetic.

−8 −6−4−2

Not Arithmetic.

Writing Rules for Arithmetic Sequences

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑

is the value of the term in position .

The first term in the sequence ()

is the index, or position of the term

The common difference

For example, the equation for the nth term of an arithmetic sequence with a first term of 3 and a common difference of 2 is given by: 𝑎𝑛=3+(𝑛−1 )2

𝑎𝑛=3+2𝑛−2

𝑎𝑛=2𝑛+1

Writing a RuleExample 2: Write a rule for the nth term of each sequence. Then find .

a) b)

55 5𝑎𝑛=𝑎1+ (𝑛−1 )𝑑𝑎𝑛=3+(𝑛−1 )5𝑎𝑛=3+5𝑛−5

𝑎𝑛=5𝑛−2

𝑎15=5 (15)−2𝑎15=73

−8−8−8

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑𝑎𝑛=55+(𝑛−1 )(−8)𝑎𝑛=55−8𝑛+8

𝑎𝑛=−8𝑛+63

𝑎15=−8 (15 )+63

𝑎15=−57

Writing a Rule Given an Term and the Common Difference

Example 3: One term of an arithmetic sequence is . The common difference is . Write a rule for the nth term. Then graph the first six terms of the sequence.

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑

𝑎19=−45 𝑑=−3

𝑎19=𝑎1+(19−1 )(−3)

−45=𝑎1+(18 )(−3)

−45=𝑎1−54

9=𝑎1

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑𝑎𝑛=9+(𝑛−1 )(−3)𝑎𝑛=9−3𝑛+3

𝑎𝑛=−3𝑛+12

1 2 3 4 5 6

9 6 3 0 -3 -6

Writing a Rule Given Two TermsExample 4: Two terms of an arithmetic sequence are and . Write a rule for the nth term.

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑𝑎7=𝑎1+ (7−1 )𝑑 17=𝑎1+6𝑑𝑎26=𝑎1+ (26−1 )𝑑 93=𝑎1+25𝑑

Subtract the second equation from the first. 76=19𝑑

Find d. 4=𝑑

Use d along with or to find . 93=𝑎1+25 (4 )93=𝑎1+100−7=𝑎1

Use and to write the general equation.

𝑎𝑛=−7+ (𝑛−1 )(4) 𝑎𝑛=−7+4𝑛−4 𝑎𝑛=4𝑛−11

Finding Sums of Finite Arithmetic Series

The expression formed by adding the terms of an arithmetic sequence is called an arithmetic series. The sum of the first n terms of a finite arithmetic series can be found using the following formula.

** notice that the sum of the terms is the average of the first and last terms multiplied by the number of terms.

𝑆𝑛=𝑛(𝑎1+𝑎𝑛2 )

Continued… 𝑆𝑛=𝑛(𝑎1+𝑎𝑛2 )

Example 5: Find the sum ∑𝑖=1

20

(3 𝑖+7 )

FIRST: Find the first and last terms:

𝑎1=3 (1 )+7

𝑎20=3 (20 )+7

¿10¿67

SECOND: Find the sum using the formula.

𝑆𝑛=𝑛(𝑎1+𝑎𝑛2 )𝑆20=20 (𝑎1+𝑎202 )𝑆20=20 ( 10+672 )𝑆20=770

Solving a Real-Life ProblemExample 6: You are making a house of cards similar to the one shown.

a) Write a rule for the number of cards in the nth row when the top row is row # 1. b) How many cards do you need to make a house of cards with 12 rows?

Row 1:Row 2:Row 3:

3 cards6 cards9 cards

𝑑=3 𝑎1=3

𝑎𝑛=𝑎1+ (𝑛−1 )𝑑𝑎𝑛=3+(𝑛−1 )3𝑎𝑛=3+3𝑛−3𝑎𝑛=3𝑛

𝑎𝑛=3𝑛𝑎12=3 (12)𝑎12=36

𝑆𝑛=𝑛(𝑎1+𝑎𝑛2 )

𝑆12=12(𝑎1+𝑎122 )𝑆12=12( 3+362 )𝑆12=234

You would need 234 cards to make a house of cards with 12 rows.

Monitoring ProgressTell whether the sequence is arithmetic, explain your reasoning.

1) 2) 3)

4) Write a rule for the nth term of the sequence then find .

Write a rule for the nth term of the sequence.

5)

6)

Arithmetic, d=3 Arithmetic, d=-6 Not arithmetic, the differences are not constant

𝑎𝑛=4𝑛+3 ;𝑎15=63

𝑎𝑛=7𝑛−27

𝑎𝑛=−5𝑛+106

Monitoring ProgressFind the sum.

7) 8) 9)

10) What if, in example 6,how many cards do you need to make a house of cards with 8 rows?

Row 1:Row 2:Row 3:

3 cards6 cards9 cards

𝑎𝑛=3𝑛

∑𝑖=1

10

9 𝑖 ∑𝑘=1

12

(7𝑘+2) ∑𝑛=1

20

(−4𝑛+6)¿ 495 ¿570 ¿−720

108𝑐𝑎𝑟𝑑𝑠