1. 5.5 Geometric Series (1/7)
In an geometric series each term increases by a constant multiplier (r)
This means the difference between consecutive terms is a common ratio (r)
Therefore: r
T11 ÷ T10 =
T37 ÷ T36 =
T317 ÷ T316 =
T11
T10 =
T37
T36 =
T317
T316 = r OR

2. 5.5 Geometric Series (2/7) T2 T1 T3 T2 m 7 847 m Example:
Find m if 7 + m is a geometric series.
Note: T2 T1 = T3 T2 m 7 =
847 m
Therefore:
m2 =
m = 77

3. 5.5 Geometric Series (3/7) T2 T1 T3 T2 Example:
Is … a geometric series?
Note: T2 T1 = T3 T2
= r
Therefore:
169 13
2028
169
= 13 and = 12
Therefore NOT Geometric Series

4. 5.5 Terms of a Geometric Series (4/7)
The terms of an geometric series are: a
+ ar
+ ar2
+ ar3
+ …
+ arn-1
+ …
Tn = a r n – 1
Value of
nth term
Common
Ratio
Term
number
1st term

5. 5.5 Terms of a Geometric Series (5/7)
Example: 3 4 6 12 12 36
Find the common ratio of + +
+ … 6 12 3 4 2 3
r = ÷ =
Find the 100th term
Tn = a r n – 1
( )
100-1 3 4 2 3
= x
3 x 299
22x399
297
398 = =

6. 5.5 Terms of a Geometric Series (6/7)
Example: …
Which term in the series above is 243.
a = 1
Tn = a rn – 1
r = 9 ÷ 3 = 3
243 = 1 x 3n – 1
Tn = 243
3n – 1 = 243
log33n – 1 = log3243
log 243
log 3
n - 1 = log3243 =
n - 1 = 5
n = 6

7. 5.5 Terms of a Geometric Series (7/7)
The 3rd term of a series is 100 and the 5th term is 2500.
Example:
Find r.
Find a.
ar5-1 =
ar4 =
2500
Tn = a rn – 1
ar3-1 =
ar2 =
100
100 = a x 53 – 1
ar4
ar2
2500
100 =
100 = a x 52
100 = a x 25
r2 = 25
a = 4
r = 5