MATHPOWERTM 12, WESTERN EDITION

1
Chapter 6 Sequences and Series
6.3 and 6.4
Geometric Sequences and Compound Interest
6.3.1
MATHPOWERTM 12, WESTERN EDITION
2
Geometric Sequences
A geometric sequence is a sequence where each
term is obtained by multiplying the preceding
term by a constant, called the common ratio.
If tn is a geometric sequence with t1 a and the
common ratio between successive terms, r, then
the general formula is
tn arn - 1

• Where
• tn is the general term of the geometric
  sequence,
• n in the position of the term being considered,
• a is the first term, and
• r is the common ratio.

You can determine r, the common ratio, for any
geometric sequence by dividing any term by the
previous term
6.3.2
3
Geometric Sequences
For the geometric sequence 4, 8, 16, 32, . .
., a) find the general term. b) find the value
of t9.
Find the common ratio
Use the general formula
Use the general term
tn arn - 1 4(2)n - 1 22(2)n -
1 22 n - 1 tn 2n 1
tn 2n 1 t9 29 1 t9 1024
6.3.3
4
Geometric Sequences
In a geometric sequence, the sixth term is 972
and the eighth term is 8748. Determine a, r, and
tn.
t6 972 972 ar5
t8 8748 8748 ar7
For r 3 972 ar5 972 a(3)5 972
243a 4 a
For r -3 972 ar5 972 a(-3)5 972
-243a -4 a
a 4 r 3 tn 4(3)n - 1 or tn
(-4)(-3)n - 1
tn arn - 1 tn 4(3)n - 1 or tn
(-4)(-3)n - 1
r2 9 r 3
6.3.4
5
Geometric Sequences - Applications
1. A photocopy machine reduces a picture to 75
of its previous size with each photocopy
taken. If it is originally 40 cm long, find
its size after the tenth reduction.
Now

1 2 3 4 5 6 7 8
9 10 11
tn arn - 1 t11 40(0.75)11 - 1 2.25
The picture will be 2.25 cm long.
2. A car that is valued at 30 000 depreciates
20 in value each year. Find its value at
the end of six years.
Now
1 2 3 4 5 6 7
tn arn - 1 t7 30 000(0.80)6 7864.32
The cars value will be 7864.32.
6.3.5
6
Geometric Sequences - Applications

• 3. At the end of the fourth year, Archbishop
  OLeary
• High School had a population of 1327 students. At
  the
• end of its tenth year, the school had 2036
  students.
• Assuming that the growth rate was consistent,
  find
• the growth rate.
• the number of students in the first year.

1327
tn arn - 1 t7 1327(r)6 2036
1327 (r)6
a)
4 5 6 7 8 9 10
The growth rate is 7.4.
b) tn arn - 1 1327 a(1.074)3
There were 1071 students in the first year.
r 1.074
a 1071
6.3.6
7
Compound Interest
The formula for compound interest is A P(1
i)n.

• Where
• A is the amount of money
• after investing a principal

• i is the rate of interest per
• compounding period

• n is the number of
• compounding periods

• P is the principal (the money
• invested or borrowed)

Example Find the accumulated amount of 3000
invested at 12 per annum for a period of five
years compounded quarterly.
A ? P 3000 i 12/a 12 4
3 n 5 x 4 20
A P(1 i)n 3000(1 0.03)20 5418.33
The amount after five years would be 5418.33.
6.4.1
8
Compound Interest
What sum invested now will amount to 10 000 in
five years at 10/a compounded semiannually?
A 10 000 P ? i 10/a 10 2
5 n 5 x 2 10
A P(1 i)n 10 000 P(1 0.05)10
P 6139.13
The initial investment would be 6139.13.
6.4.2
9
Assignment
Suggested Questions Pages 300 and 301 1-21 odd,
23 a, 24,27, 28, 30, 34
Pages 304 and 305 17-27 odd, 28, 30
6.4.3