MCR 3U Grade 11 University
Geometric Sequences
Example 1: Doubling
E. coli bacteria live in the intestine and help your body to absorb certain
vitamins. However, one strain of E. coli bacteria is particularly harmfull. The
rate at which bacteria cells divide and reproduce determines the time it takes for
the bacteria to affect your health. E. coli bacteria are very small. Anout 50 cells
lined up end to end would be roughly a thick as a strand of your hair. As few as
100 cells would cause infection. Each bacterium divides in two about every onehalf hour to one hour.
1. Assume that 100 E. coli bacteria double in number every hour. Copy and
complete the table to show how many bacteria are present after each hour for
8 h.
Time (h)
0
1
2
3
4
5
6
7
8
Number of 100 200 400 800 1600 3200 6400 12800 25600
Bacteria
2. Write the numbers of bacteria after each hour for 8 h as a sequence.
100, 200, 400, 800, 1600, 3200, 6400, 12800, 25800
3. Is there a common difference?
No: 200 – 100 = 100,
400 – 200 = 200,
800 – 400 = 400,
........................
4. Graph this sequence. Describe the rate of change between pairs of
consecutive terms.
The rate of change between pairs of consecutive terms increases.
5. Divide each term; except the first, by previous term. What do you notice?
What is the significance of this value?
𝟐𝟎𝟎
𝟏𝟎𝟎
=
𝟒𝟎𝟎
𝟐𝟎𝟎
=
𝟖𝟎𝟎
𝟒𝟎𝟎
=⋯=
𝟐𝟓𝟔𝟎𝟎
𝟏𝟐𝟖𝟎𝟎
= 𝟐.
6. Rewrite each term of the sequence that you wrote in step 2 as an expression
of the original 100 bacteria and the value that you calculated in step 5.
𝒕𝟐 = 𝟏𝟎𝟎 × 𝟐, 𝒕𝟑 = 𝟏𝟎𝟎 × 𝟐𝟐 , 𝒕𝟒 = 𝟏𝟎𝟎 × 𝟐𝟑 , 𝒕𝟓 = 𝟏𝟎𝟎 × 𝟐𝟒 ,
𝒕𝟔 = 𝟏𝟎𝟎 × 𝟐𝟓 , 𝒕𝟕 = 𝟏𝟎𝟎 × 𝟐𝟔 , 𝒕𝟖 = 𝟏𝟎𝟎 × 𝟐𝟕 , 𝒕𝟗 = 𝟏𝟎𝟎 × 𝟐𝟖 .
7. Write the general term for this sequence.
𝒕𝒏 = 𝟏𝟎𝟎 × 𝟐𝒏 − 𝟏 , 𝒏 ∈ ℕ .
8. Determine the number of bacteria after 24 h.
𝒕𝟐𝟓 = 𝟏𝟎𝟎 × 𝟐𝟐𝟒
Geometric Sequence
In a geometric sequence, the ratio of any term, except the first one, to the
previous term, is constant for all pairs of consecutive terms. This constant is
referred to as to common ratio r.
For example,
4, 12, 36, 108 ... – is an increasing geometric sequence with r = 3;
𝟏
432, 144, 48, 16... – is a decreasing geometric sequence with r = .
𝟑
Example 2: Half-Life
Half-life is the time required for a radioactive material to decay to one-half of its
original mass. Carbon-11, used in medical applications, has a half-life of only
20 min.
1. After a medical procedure, Harry’s body has about 100 mg of carbon-11.
Complete the table below to show the mass of carbon-11 after each 20-min
interval for 3 h.
20-min
Interval
Mass of
Carbon11 (mg)
0
1
2
3
4
5
6
7
8
9
100
50
25
12.5
6.25
3.13
1.56
0.78
0.39
0.20
2. Write the mass of carbon-11 after each 20-min interval for 3 h as a sequence.
100, 50, 25, 12.5, 6.25, 3.13, 1.56, 0.78, 0.39, 0.20
3. Write the general term for this sequence.
𝟏 𝒏− 𝟏
𝒕𝒏 = 𝟏𝟎𝟎 × ( )
𝟐
, 𝒏 ∈ ℕ.
4. Determine the mass of carbon-11 remaining after 24 h.
24 hours = 48 × 30 min
𝟏 𝟒𝟖
𝒕𝟒𝟗 = 𝟏𝟎𝟎 × ( )
𝟐
≈ ???