1. Chapter 13: Sequences and Series
Section 13.1: Arithmetic and Geometric Sequences

2. arithmetic sequence – a sequence of numbers such that the difference of any two consecutive terms is constant.
tn = t (n – 1)d
To get the start with and add the
nth term the first difference
term n – 1 times

3. geometric sequence – a sequence such that the ratio (common ratio) of any two consecutive terms is constant.
tn = t1 · r (n – 1)
To get the start with and multiply
nth term the first by the ratio
term n – 1 times

4. sequence – defined to be a function whose domain is the set of positive integers.

5. Example 1:
State whether the given sequence is
arithmetic, geometric, or neither.
1 · 2, 2 · 3, 3 · 4, 4 · 5, …
Neither the difference of consecutive terms nor the ratio of consecutive terms is constant.
Neither

6. 51/3 , 101/3 , 201/3, 401/3, …
The common ratio of the terms is 21/3.
geometric
15, 11, 7, 3, …
The common difference is -4.
Arithmetic

7. Example 2:
State the first four terms of the specified
sequence. Then tell whether the sequence
is arithmetic, geometric, or neither.
tn = 5n + 2
= 5(1) + 2
= 5(2) + 2
= 5(3) + 2
= 5(4) + 2
7, 12, 17, 22, …
arithmetic

8. tn = n + 1
n + 2
tn = 3n
tn = n3
b) 2/3, ¾, 4/5, 5/6, … (neither)
3, 9, 27, 81, … (geometric)
1, 8, 27, 64, …(neither)

9. Example 3:
Find a formula for tn.
1, 4, 7, 10, …
b) 8, 4, 2, 1, …

10. HOMEWORK (Day 1)
pg. 476; 1 – 8, 13, 15

11. Example 4: a. Arithmetic sequence t1 = 114, t3 = 100, Find t20
Example 4: a. Arithmetic sequence t1 = 114, t3 = 100, Find t = (3 – 1)d -14 = 2d -7 = d t20 = (20 – 1)(-7) t20 = -19

12. Geometric sequence
t1 = 5/64, t4 = 5/8, find t26.
5/8 = 5/64(r)(4 – 1)
8 = r3
r = 2
t26 = 5/64(2)(26-1)
t26 = 2,621,440

13. Practice Arithmetic Sequence t1 = 5, t7 = 29, find t53. d = 4
Geometric Sequence
t1 = ½ , t5 = 8, find t10.
r = 2
t10 = 256

14. HOMEWORK (Day 2)
pg. 477; 29, 30, 33, 34, 36

15. Example 5:
In a certain sequence, t2 = 2 and t5 = 16.
Find t10 if the sequence is:
Arithmetic sequence
t2 = 2
2 = t1 + (2 – 1)d
2 = t1 + d

16. t5 = 16
16 = t1 + (5 – 1)d
16 = t1 + 4d
(2 = t1 + d)
14 = 3d
d = 14/3

17. 2 = t1 + d
t1 = 2 – d
t1 = 2 – 14/3
t1 = -8/3
t10 = t1 + (10 – 1)d
= -8/3 + (9)(14/3)
= 118 3

18. Geometric sequence
t2 = 2
2 = t1 · r2 – 1
2 = t1 · r
t5 = 16
16 = t1 · r5 – 1
16 = t1 · r4
t1 = 2 r
t1 = 16 r4

19. 2 = 16
r r4
2r4 = 16r
r3 = 8
r = 2
t1 = 2/r
= 2/2
= 1
t10 = t1 · r10 – 1
= 1 · r10 – 1
= 1 · 29
= 512

20. HOMEWORK (Day 3)
pg. 477; 31, 32, 35

21. Arithmetic; t1 = 3, t5 = 23, find t20.
Geometric; t1 = 24, t6 = ¾ , find t10.
Arithmetic; t2 = 5, t8 = 23, find t30.
Geometric; t4 = 32, t10 = 2,048, find t25.