1. SECTIONS 9-2 and 9-3 : ARITHMETIC &
ALGEBRA II HONORS/GIFTED : SECTIONS 9-2 and 9-3 (Arithmetic and Geometric Sequences)
ALGEBRA II H/G @
SECTIONS 9-2 and 9-3 : ARITHMETIC &
GEOMETRIC SEQUENCES

2. What are the next three terms in each sequence?
1) 3, 12, 21, _______, _______, _______
SOLUTION : 30, 39, 48. You add 9 to get the next term.
2) 3, 12, 48, _______, _______, _______
SOLUTION : 192, 768, You multiply by 4 to get the next term.
3) 3, 12, 72, _______, _______, _______
SOLUTION : I don’t know either. There is no pattern.

3. PROGRESSING TO INFINITY
ARITHMETIC SEQUENCE : A progression in which one term equals a constant (common difference) added to the preceding term.
GEOMETRIC SEQUENCE : A progression in which one term equals a constant (ratio) multiplied by the preceding term.

4. Determine whether each sequence is arithmetic, geometric, or neither
Determine whether each sequence is arithmetic, geometric, or neither. Explain your answer.
4) 4, 7, 10, …
SOLUTION : Arithmetic. The common difference is 3.
5) 2, 6, 24, …
SOLUTION : Neither. There is neither a common difference nor a ratio.
6) 3, -6, 12, …
SOLUTION : Geometric. The ratio is -2.

5. Given the arithmetic sequence :
a1 a2 a3 a4 a5 a6 …
NOTE : a1 means 1st term, a2 the 2nd term, and so on.
7) What is the common difference?
SOLUTION : 7
a1 = 3 + 7(0) a4 = 3 + 7(3)
a2 = 3 + 7(1) a5 = 3 + 7(4)
a3 = 3 + 7(2) a6 = 3 + 7(5)
8) an = ___________________
SOLUTION : an = a1 + (n – 1)d

6. ARITHMETIC SEQUENCE FORMULA
an = a1 + (n – 1)d
RED means the formula is to be written on your formulas page.
an stands for the nth term
a1 stands for the first term
n is the term number
d is the common difference

7. Given the geometric sequence :
a1 a2 a3 a4 a5 a6 …
9) What is the ratio?
SOLUTION : 2
a1 = 3 • 20 a4 = 3 • 23
a2 = 3 • a5 = 3 • 24
a3 = 3 • a6 = 3 • 25
10) an = ___________________
SOLUTION : an = a1 • rn-1

8. GEOMETRIC SEQUENCE FORMULA
an = a1 • rn-1
RED means the formula is to be written on your formulas page.
an stands for the nth term
a1 stands for the first term
n is the term number
r is the ratio

9. 11) Calculate a100 for the arithmetic sequence
11, 16, 21, 26, …
SOLUTION : an = a1 + (n – 1)d
a100 = 11 + (100 – 1)5
= 11 + (99)5 =
= 506
Therefore, the 100th term is 506

10. 12) Calculate a100 for the geometric sequence with
a1 = 40 and r = 1.05.
SOLUTION : an = a1 • rn-1
a100 = 40 • =
Therefore, the 100th term of the sequence is

11. 13) The number 68 is a term in the arithmetic sequence with a1 = 5 and d = 3. Which term is it?
SOLUTION : an = a1 + (n – 1)d
68 = 5 + (n – 1)3
68 = 5 + 3n – 3
68 = 3n + 2
66 = 3n
22 = n
Therefore, 68 is the 22nd term.

12. 14) A geometric sequence has a1 = 17 and r = 2. If
an = 34816, find n.
SOLUTION : an = a1 • rn-1
34816 = 17 • 2n-1
2048 = 2n-1
log2048 = (n – 1)log2
log2048 = n – 1
log 2
= n – 1
= n
Therefore, there are 12 terms in the sequence.

13. 15) Calculate a45 for the arithmetic sequence
100, 94, 88, 82,…
SOLUTION : an = a1 + (n – 1)d
a45 = (45 – 1)(-6)
= (44)(-6)
= (-264)
= -164
Therefore, the 45th term is -164.

14. MEAN : average
ARITHMETIC or GEOMETRIC MEANS :
between two numbers are the terms which form an arithmetic sequence or a geometric sequence between the two given terms.

15. 16) Insert 4 arithmetic means between 37 and 52.
SOLUTION :
37, _______, _______, _______, _______, 52
an = a1 + (n – 1)d = 40
= 37 + (6 – 1)d = 43
52 = d = 46
15 = 5d = 49
3 = d

16. 17) Insert 2 geometric means between 52 and 73.
SOLUTION :
52, _______, _______, 73
an = a1 • rn • 1.12 = 58.24
= 52 • r • 1.12 = 65.23
73 = 52 • r3
= r3
1.12 = r

17. 18) Find a1 for the arithmetic sequence with
a19 = 50 and a20 = 53.
SOLUTION : an = a1 + (n – 1)d
53 = a1 + (20 – 1)(3)
53 = a1 + (19)(3)
53 = a1 + 57
-4 = a1
Therefore, the 1st term of the sequence is -4.

18. 19) Calculate a1 for the geometric sequence with
a9 = 200 and a10 = 220.
SOLUTION : an = a1 • rn-1
220 = a1 • (1.1)10-1
220 = a1
1.19
93.3 = a1
Therefore, the 1st term of the sequence is 93.3.