This document provides information about geometric and arithmetic sequences. It defines geometric sequences as sequences where the ratio of successive terms is always the same number, called the common ratio. Arithmetic sequences are defined as sequences where each term is calculated by adding the same number, called the common difference, to the previous term. Several examples of determining common ratios and differences are provided, as well as formulas for calculating terms in geometric and arithmetic sequences.

1. GEOMETRIC SEQUENCE AND
ARITHMATIC SEQUENCE
GRADE 12
MATHEMATICS
BY
MR MADONSELA NR

2. GEOMETRIC
SEQUENCES
These are sequences where the ratio of
successive terms of a sequence is always
the same number. This number is called
the common ratio.
GRADE 12

3. An introduction…………
1, 4, 7,10,13
9,1, 7, 15
6.2, 6.6, 7, 7.4
, 3, 6
− −
π π + π +
Arithmetic Sequences
ADD
To get next term
2, 4, 8,16, 32
9, 3,1, 1/3
1,1/ 4,1/16,1/ 64
, 2.5 , 6.25
− −
π π π
Geometric Sequences
MULTIPLY
To get next term
d = 3
d = −8
d = 0.4
d = 3
r = 2
r = −
1
3
r = 1
4
r = 2.5

4. Ex: Determine if the sequence is
geometric. If so, identify the
common ratio
 1, -6, 36, -216
yes. Common ratio=-6
 2, 4, 6, 8
no. No common ratio
This is an Arithmetic Sequence with “common difference” of 2

5. Important Formula for
Geometric Sequence:
an = a1  r n-1
Where:
an is the nth term in the sequence
a1 is the first term
n is the number of the term
r is the common ratio

6. Ex: Write the first 4 terms of
this sequence with:
First term: a1 = 7
Common ratio = 1/3
an = a1 * r n-1
Now find the first five terms:
a1 = 7(1/3) (1-1)
= 7
a2 = 7(1/3) (2-1)
= 7/3
a3 = 7(1/3) (3-1)
= 7/9
a4 = 7(1/3) (4-1)
= 7/27
a5 = 7(1/3) (5-1)
= 7/81

7. Geometric Sequence Problem
Find the 19th
term in the sequence of 11,33,99,297 . . .
a19 = 11(3)18
=4,261,626,379
Common ratio = 3
a19 = 11 (3) (19-1)
Start with the sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1

8. Let’s try one
Find the 10th
term in the sequence of
1, -6, 36, -216 . . .
a10 = 1(-6)9
= -10,077,696
Common ratio = -6
a10 = 1 (-6) (10-1)
Start with the sequence formula
Find the common ratio
between the values.
Plug in known values
Simplify
an = a1 * r n-1

9. × 2 × 2 × 2 × 2
r = 2
1−
= n
n ara
Try this to get the 5th term.
a = 1
( ) 1621
15
5 ==
−
a
1, 2, 4, 8, 16 . . .

10. Find the 8th term of 0.4, 0.04. 0.004, . . .
1−
= n
n ara
1.0
4.0
04.0
==r
To find the common ratio, take any
term and divide it by the term in
front
( ) 1
1.04.0
−
=
n
na
( ) 00000004.01.04.0
18
8 ==
−
a

11. Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
2 9 5 2 7− − − = − − =
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33

12. Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of x, 2x, 3x, …
Arithmetic Sequence, d = x
4x, 5x, 6x, 7x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -32k

13. Given an arithmetic sequence with 15 1a 38 and d 3, find a .= = −
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
d common difference→
x
15
38
NA
-3
( )n 1a a n 1 d= + −
( ) ( )38 x 1 15 3= + − −
X = 80

14. 16 1Find a if a 1.5 and d 0.5= =Try this one:
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
d common difference→
1.5
16
x
NA
0.5
( )n 1a a n 1 d= + −
( )16 1.5 0.a 16 51= + −
16a 9=

15. n 1Find n if a 633, a 9, and d 24= = =
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
d common difference→
9
x
633
NA
24
( )n 1a a n 1 d= + −
( )633 9 21x 4= + −
633 9 2 244x= + −
X = 27

16. 1 29Find d if a 6 and a 20= − =
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
d common difference→
-6
29
20
NA
x
( )n 1a a n 1 d= + −
( )120 6 29 x= + −−
26 28x=
13
x
14
=

17. Example 7. An auditorium has 20 rows of seats. There are 20 seats in
the first row, 21 seats in the second row, 22 seats in the third row, and
so on. How many seats are there in all 20 rows?
1 20 1 19d c= = − =
( ) ( )1 201 20 19 1 39na a n d a= + − → = + =
( ) ( )20
20
20 39 10 59 590
2
S = + = =

18. Example 8. A small business sells $10,000 worth of sports memorabilia
during its first year. The owner of the business has set a goal of
increasing annual sales by $7500 each year for 19 years. Assuming that
the goal is met, find the total sales during the first 20 years this business
is in operation.
1 10,000 7500 10,000 7500 2500a d c= = = − =
( ) ( )1 201 10,000 19 7500 152,500na a n d a= + − → = + =
( ) ( )20
20
10,000 152,500 10 162,500 1,625,000
2
S = + = =
So the total sales for the first 2o years is $1,625,000

19. Find the next three terms of 2, 3, 9/2, ___, ___, ___
3 – 2 vs. 9/2 – 3… not arithmetic
3 9/ 2 3
1.5 geometric r
2 3 2
= = → → =
3 3 3 3 3 3
2 2 2
9
2, 3, , , ,
2
9 9 9
2 2 2 2 2 2
× × × × × ×
9
2, 3, , ,
27 81 243
4 8
,
2 16

20. 1 9
1 2
If a ,r , find a .
2 3
= =
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
r common ratio→
1/2
x
9
NA
2/3
n 1
n 1a a r −
=
9 1
1 2
x
2 3
−
  
=   
  
8
8
2
x
2 3
=
×
7
8
2
3
=
128
6561
=

21. 9Find a of 2, 2, 2 2,...
1a First term→
na nth term→
nS sum of n terms→
n number of terms→
r common ratio→
x
9
NA
2
2 2 2
r 2
22
= = =
n 1
n 1a a r −
=
( )
9 1
x 2 2
−
=
( )
8
x 2 2=
x 16 2=