1. Splash Screen

2. Concept

3. 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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

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

14. Concept

15. 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

16. 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

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

18. 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

19. 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

20. 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

21. 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

22. Homework P. 670 # 3-12 (x3), 16 – 52 (x4) skip 32

23. End of the Lesson