1. Splash Screen

2. Five-Minute Check (over Lesson 10–1) CCSS Then/Now New Vocabulary
Key Concept: nth Term of an Arithmetic Sequence
Example 1: Find the nth term
Example 2: Write Equations for the nth Term
Example 3: Find Arithmetic Means
Key Concept: Partial Sum of an Arithmetic Series
Example 4: Use the Sum Formulas
Example 5: Find the First Three Terms
Key Concept: Sigma Notation
Example 6: Standardized Test Example: Use Sigma Notation
Lesson Menu

3. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 1

4. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 1

5. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 2

6. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 2

7. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 3

8. Determine whether the sequence is arithmetic, geometric, or neither
A. arithmetic
B. geometric
C. neither
5-Minute Check 3

9. Find the next three terms of the sequence. 25, 50, 75, 100, …
A. 125, 150, 175
B. 125, 250, 500
C. 125, 145, 175
D. 150, 200, 225
5-Minute Check 4

10. Find the next three terms of the sequence. 25, 50, 75, 100, …
A. 125, 150, 175
B. 125, 250, 500
C. 125, 145, 175
D. 150, 200, 225
5-Minute Check 4

11. Find the next three terms of the sequence. –1, –6, –36, –216, …
A. –236, –266, –336
B. –306, –336, –416
C. –1296, –7776, –46,656
D. –1296, –3888, –11,664
5-Minute Check 5

12. Find the next three terms of the sequence. –1, –6, –36, –216, …
A. –236, –266, –336
B. –306, –336, –416
C. –1296, –7776, –46,656
D. –1296, –3888, –11,664
5-Minute Check 5

13. Find the first term and the ninth term of the arithmetic sequence
B. 2.5; 22
C. 2; 22
D. 2.5; 14.5
5-Minute Check 6

14. Find the first term and the ninth term of the arithmetic sequence
B. 2.5; 22
C. 2; 22
D. 2.5; 14.5
5-Minute Check 6

15. Mathematical Practices
Content Standards
A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
Mathematical Practices
8 Look for and express regularity in repeated reasoning.
CCSS

16. You determined whether a sequence was arithmetic.
Use arithmetic sequences.
Find sums of arithmetic series.
Then/Now

17. arithmetic means series arithmetic series partial sum sigma notation
Vocabulary

18. Concept

19. Find the 20th term of the arithmetic sequence 3, 10, 17, 24, … .
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.
Example 1

20. an = a1 + (n – 1)d nth term of an arithmetic sequence
Find the nth Term
Step 2 Find the 20th term.
an = a1 + (n – 1)d nth term of an arithmetic sequence
a20 = 3 + (20 – 1)7 a1 = 3, d = 7, n = 20
= or 136 Simplify.
Answer:
Example 1

21. an = a1 + (n – 1)d nth term of an arithmetic sequence
Find the nth Term
Step 2 Find the 20th term.
an = a1 + (n – 1)d nth term of an arithmetic sequence
a20 = 3 + (20 – 1)7 a1 = 3, d = 7, n = 20
= or 136 Simplify.
Answer: The 20th term of the sequence is 136.
Example 1

22. Find the 17th term of the arithmetic sequence 6, 14, 22, 30, … .
B. 140
C. 146
D. 152
Example 1

23. Find the 17th term of the arithmetic sequence 6, 14, 22, 30, … .
B. 140
C. 146
D. 152
Example 1

24. d = –6 – (–8) or 2; –8 is the first term.
Write Equations for the nth Term
A. Write an equation for the nth term of the arithmetic sequence below. –8, –6, –4, …
d = –6 – (–8) or 2; –8 is the first term.
an = a1 + (n – 1)d nth term of an arithmetic sequence
an = –8 + (n – 1)2 a1 = –8 and d = 2
an = –8 + (2n – 2) Distributive Property
an = 2n – 10 Simplify.
Answer:
Example 2A

25. d = –6 – (–8) or 2; –8 is the first term.
Write Equations for the nth Term
A. Write an equation for the nth term of the arithmetic sequence below. –8, –6, –4, …
d = –6 – (–8) or 2; –8 is the first term.
an = a1 + (n – 1)d nth term of an arithmetic sequence
an = –8 + (n – 1)2 a1 = –8 and d = 2
an = –8 + (2n – 2) Distributive Property
an = 2n – 10 Simplify.
Answer: an = 2n – 10
Example 2A

26. an = a1 + (n – 1)d nth term of an arithmetic sequence
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)d nth term of an arithmetic sequence
11 = a1 + (6 – 1)(–11) a6 = 11, n = 6, and d = –11
11 = a1 – 55 Multiply.
66 = a1 Add 55 to each side.
Example 2B

27. Then write the equation.
Write Equations for the nth Term
Then write the equation.
an = a1 + (n – 1)d nth term of an arithmetic sequence
an = 66 + (n – 1)(–11) a1 = 66, and d = –11
an = 66 + (–11n + 11) Distributive Property
an = –11n + 77 Simplify.
Answer:
Example 2B

28. Then write the equation.
Write Equations for the nth Term
Then write the equation.
an = a1 + (n – 1)d nth term of an arithmetic sequence
an = 66 + (n – 1)(–11) a1 = 66, and d = –11
an = 66 + (–11n + 11) Distributive Property
an = –11n + 77 Simplify.
Answer: an = –11n + 77
Example 2B

29. A. Write an equation for the nth term of the arithmetic sequence below
A. an = –9n – 21
B. an = 9n – 21
C. an = 9n + 21
D. an = –9n + 21
Example 2A

30. A. Write an equation for the nth term of the arithmetic sequence below
A. an = –9n – 21
B. an = 9n – 21
C. an = 9n + 21
D. an = –9n + 21
Example 2A

31. B. Write an equation for the nth term of the arithmetic sequence below
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
Example 2B

32. B. Write an equation for the nth term of the arithmetic sequence below
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
Example 2B

33. Find the arithmetic means in the sequence 21, ___, ___, ___, 45, … .
Find Arithmetic Means
Find the arithmetic means in the sequence 21, ___, ___, ___, 45, … .
Step 1 Since there are three terms between the first and last terms given, there are or 5 total terms, so n = 5.
Step 2 Find d.
an = a1 + (n – 1)d Formula for the nth term
45 = 21 + (5 – 1)d n = 5, a1 = 21, a5 = 45
45 = d Distributive Property
24 = 4d Subtract 21 from each side.
6 = d Divide each side by 4.
Example 3

34. Step 3 Use the value of d to find the three arithmetic means.
Find Arithmetic Means
Step 3 Use the value of d to find the three arithmetic means. +6
Answer:
Example 3

35. Step 3 Use the value of d to find the three arithmetic means.
Find Arithmetic Means
Step 3 Use the value of d to find the three arithmetic means. +6
Answer: The arithmetic means are 27, 33, and 39.
Example 3

36. Find the three arithmetic means between 13 and 25.
Example 3

37. Find the three arithmetic means between 13 and 25.
Example 3

38. Concept

39. We need to find n before we can use one of the formulas.
Use the Sum Formulas
Find the sum … + 80.
Step 1 a1 = 8, an = 80, and d = 12 – 8 or 4.
We need to find n before we can use one of the formulas.
an = a1 + (n – 1)d nth term of an arithmetic sequence
80 = 8 + (n – 1)(4) an = 80, a1 = 8, and d = 4
80 = 4n + 4 Simplify.
19 = n Solve for n.
Example 4

40. Step 2 Use either formula to find Sn.
Use the Sum Formulas
Step 2 Use either formula to find Sn.
Sum formula
a1 = 8, n = 19, d = 4
Simplify.
Answer:
Example 4

41. Step 2 Use either formula to find Sn.
Use the Sum Formulas
Step 2 Use either formula to find Sn.
Sum formula
a1 = 8, n = 19, d = 4
Simplify.
Answer: 836
Example 4

42. Find the sum … + 68.
A. 318
B. 327
C. 340
D. 365
Example 4

43. Find the sum … + 68.
A. 318
B. 327
C. 340
D. 365
Example 4

44. Step 1 Since you know a1, an, and Sn, use to find n.
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.
Example 5

45. an = a1 + (n – 1)d nth term of an arithmetic sequence
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 each side.
3 = d Divide each side by 5.
Example 5

46. Step 3 Use d to determine a2 and a3. a2 = 14 + 3 or 17
Find the First Three Terms
Step 3 Use d to determine a2 and a3.
a2 = or 17
a3 = or 20
Answer:
Example 5

47. Step 3 Use d to determine a2 and a3. a2 = 14 + 3 or 17
Find the First Three Terms
Step 3 Use d to determine a2 and a3.
a2 = or 17
a3 = or 20
Answer: The first three terms are 14, 17, and 20.
Example 5

48. Find the first three terms of an arithmetic series in which a1 = 11, an = 31, and Sn = 105.
B. 11, 16, 21
C. 11, 17, 23, 30
D. 17, 23, 30, 36
Example 5

49. Find the first three terms of an arithmetic series in which a1 = 11, an = 31, and Sn = 105.
B. 11, 16, 21
C. 11, 17, 23, 30
D. 17, 23, 30, 36
Example 5

50. Concept

51. You need to find the sum of the series. Find a1, an, and n.
Use Sigma Notation
Evaluate
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.
Example 6

52. 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) or 21
Example 6

53. Method 2 Find the terms by replacing k with 3, 4, ..., 10. Then add.
Use Sigma Notation
Solve the Test Item
Method 2 Find the terms by replacing k with 3, 4, ..., 10. Then add.
Example 6

54. Use Sigma Notation
Answer:
Example 6

55. Answer: The sum of the series is 112. The correct answer is D.
Use Sigma Notation
Answer: The sum of the series is 112. The correct answer is D.
Example 6

56. Evaluate
A. 85
B. 95
C. 108
D. 133
Example 6

57. Evaluate
A. 85
B. 95
C. 108
D. 133
Example 6

58. End of the Lesson