Five-Minute Check (over Lesson 10–1)CCSSThen/NowNew VocabularyKey Concept: nth Term of an Arithmetic SequenceExample 1: Find the nth termExample 2: Write Equations for the nth TermExample 3: Find Arithmetic MeansKey Concept: Partial Sum of an Arithmetic SeriesExample 4: Use the Sum FormulasExample 5: Find the First Three TermsKey Concept: Sigma NotationExample 6: Standardized Test Example: Use Sigma Notation

Over Lesson 10–1

A. arithmetic

B. geometric

C. neither

Determine whether the sequence is arithmetic, geometric, or neither.18, 11, 4, …

Over Lesson 10–1

A. arithmetic

B. geometric

C. neither

Determine whether the sequence is arithmetic, geometric, or neither.1, –2, 4, –8, …

Over Lesson 10–1

A. arithmetic

B. geometric

C. neither

Determine whether the sequence is arithmetic, geometric, or neither.5, 6, 8, 11, …

Over Lesson 10–1

A. 125, 150, 175

B. 125, 250, 500

C. 125, 145, 175

D. 150, 200, 225

Find the next three terms of the sequence.25, 50, 75, 100, …

Over Lesson 10–1

A. –236, –266, –336

B. –306, –336, –416

C. –1296, –7776, –46,656

D. –1296, –3888, –11,664

Find the next three terms of the sequence.–1, –6, –36, –216, …

Over Lesson 10–1

A. 2; 14.5

B. 2.5; 22

C. 2; 22

D. 2.5; 14.5

Find the first term and the ninth term of the arithmetic sequence.___, 4.5, 7, 9.5, 12, …

Content StandardsA.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.Mathematical Practices8 Look for and express regularity in repeated

reasoning.

You determined whether a sequence was arithmetic.

• Use arithmetic sequences.

• Find sums of arithmetic series.

• arithmetic means

• series

• arithmetic series

• partial sum

• sigma notation

Find the nth Term

Find the 20th term of the arithmetic sequence 3, 10, 17, 24, … .

Step 1 Find the common difference.

24 – 17 = 7 17 – 10 = 7 10 – 3 = 7

So, d = 7.

Find the nth Term

Answer: The 20th term of the sequence is 136.

Step 2 Find the 20th term.

an = a1 + (n – 1)d nth term of anarithmetic

sequence

a20 = 3 + (20 – 1)7 a1 = 3, d = 7, n = 20

= 3 + 133 or 136 Simplify.

A. 134

B. 140

C. 146

D. 152

Find the 17th term of the arithmetic sequence 6, 14, 22, 30, … .

Write Equations for the nth Term

A. Write an equation for the nth term of the arithmetic sequence below.–8, –6, –4, …

Answer: an = 2n – 10

d = –6 – (–8) or 2; –8 is the first term.

an = a1 + (n – 1)dnth term of an arithmetic sequence

an = –8 + (n – 1)2a1 = –8 and d = 2

an = –8 + (2n – 2)Distributive Property

an = 2n – 10Simplify.

Write Equations for the nth Term

B. Write an equation for the nth term of the arithmetic sequence below.a6 = 11, d = –11

First, find a1.

an = a1 + (n – 1)dnth term of an arithmetic sequence

11 = a1 + (6 – 1)(–11)a6 = 11, n = 6, and d = –11

11 = a1 – 55Multiply.

66 = a1Add 55 to each side.

Write Equations for the nth Term

Answer: an = –11n + 77

Then write the equation.

an = a1 + (n – 1)dnth term of an arithmetic sequence

an = 66 + (n – 1)(–11)a1 = 66, and d = –11

an = 66 + (–11n + 11)Distributive Property

an = –11n + 77Simplify.

A. Write an equation for the nth term of the arithmetic sequence below.–12, –3, 6, …

A. an = –9n – 21

B. an = 9n – 21

C. an = 9n + 21

D. an = –9n + 21

B. Write an equation for the nth term of the arithmetic sequence below.a4 = 45, d = 5

A. an = 5n + 25

B. an = 5n – 20

C. an = 5n + 40

D. an = 5n + 30

Find Arithmetic Means

Find the arithmetic means in the sequence21, ___, ___, ___, 45, … .Step 1 Since there are three terms between the first

and last terms given, there are 3 + 2 or 5total terms, so n = 5.

Step 2 Find d.

an = a1 + (n – 1)dFormula for the nth term

45 = 21 + (5 – 1)dn = 5, a1 = 21, a5 = 45

45 = 21 + 4d Distributive Property24 = 4dSubtract 21 from each side.

6 = d Divide each side by 4.

Find Arithmetic Means

Step 3 Use the value of d to find the threearithmetic means.

Answer: The arithmetic means are 27, 33, and 39.

21 27 33 39 45

+6 +6 +6 +6

A. 16, 19, 22

B. 17, 21, 25

C. 13, 17, 21

D. 15, 17, 19

Find the three arithmetic means between 13 and 25.

Use the Sum Formulas

Find the sum 8 + 12 + 16 + … + 80.Step 1 a1 = 8, an = 80, and d = 12 – 8 or 4.

We need to find n before we can use one ofthe formulas.

an = a1 + (n – 1)d nth term of anarithmetic sequence

80 = 8 + (n – 1)(4) an = 80, a1 = 8,and d = 4

80 = 4n + 4 Simplify.19 = n Solve for n.

Use the Sum Formulas

Step 2 Use either formula to find Sn.

Answer: 836

Sum formula

a1 = 8, n = 19, d = 4

Simplify.

A. 318

B. 327

C. 340

D. 365

Find the sum 5 + 12 + 19 + … + 68.

Find the First Three Terms

Find the first three terms of an arithmetic series in which a1 = 14, an = 29, and Sn = 129.Step 1 Since you know a1, an, and Sn, use

to find n.

Sum formula

Sn = 129, a1 = 14, an = 29

Simplify.

Divide each side by 43.

Find the First Three Terms

Step 2 Find d.

an = a1 + (n – 1)d nth term of an arithmetic

sequence

29 = 14 + (6 – 1)d an = 29, a1 = 14, n = 6

15 = 5d Subtract 14 from eachside.

3 = d Divide each side by 5.

Find the First Three Terms

Step 3 Use d to determine a2 and a3.

a2 = 14 + 3 or 17

a3 = 17 + 3 or 20

Answer: The first three terms are 14, 17, and 20.

A. 16, 21, 26

B. 11, 16, 21

C. 11, 17, 23, 30

D. 17, 23, 30, 36

Find the first three terms of an arithmetic series in which a1 = 11, an = 31, and Sn = 105.

A. 23 B. 70

C. 98 D. 112

Read the Test Item

You need to find the sum of the series. Find a1, an, and n.

Evaluate .

Use Sigma Notation

Method 1 Since the sum is an arithmetic series, use

the formula . There are 8

terms.

a1 = 2(3) + 1 or 7, and a8 = 2(10) + 1

or 21

Use Sigma Notation

Solve the Test ItemMethod 2 Find the terms by replacing k with

3, 4, ..., 10. Then add.

Use Sigma Notation

Answer: The sum of the series is 112. The correct answer is D.

Use Sigma Notation

A. 85

B. 95

C. 108

D. 133

Evaluate .