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Geometric Sequences — Zombiepocalypse
Summary. When last we met, Zombies were taking over the world. We started Day Z with exactly one
zombie. On each subsequent day, every zombie in existence turned exactly one human into a new
zombie. The table below summarizes your findings.
Day
Zombies
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Day
1
2
4
8
16
32
64
128
256
512
1,024
2,048
4,096
8,192
16,384
32,768
65,536
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
Zombies
131,072
262,144
524,288
1,048,576
2,097,152
4,194,304
8,388,608
16,777,216
33,554,432
67,108,864
134,217,728
268,435,456
536,870,912
1,073,741,824
2,147,483,648
4,294,967,296
8,589,934,592
Sequences. We used a table to show our data. Another way to write this is as a sequence:
Zombies = {
⏟
1
, 2, 4, 8,
1st term
16
⏟
, 32, 64, 128, 256,
5th term
512
⏟
, 1024, 2048, 4096, 8192, 16384
⏟
,…}
15th term
10th term
Sequence Notation. Usually, mathematicians will use a single letter to denote their sequence. Let’s
call ours 𝑍, for Zombies. We’re also going to use a subscript to refer to our terms:
𝑍𝑛 = {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384}
Now, when we say 𝑍𝑛 what we mean is “Tell me what the 𝑛th term of 𝑍 is.” For example, 𝑍10 = 512.
Practice.
1. 𝑍5 =
2. 𝑍17 =
3. 𝑍30 =
4. If 𝑍𝑛 = 64, what is
the value of 𝑛?
5. If 𝑍𝑛 = 8,192, what is
the value of 𝑛?
6. How many total
zombies are there
on day 10?
Defining a Sequence using a Recursive Equation. Sometimes, it makes sense to write out an
equation for your sequence. This makes it easier to find values and to share your work with others. A
recursive equation gives you a way to find the next value of a sequence if you already know the one
before it.
You already found out that the terms of our sequence double every time. In other words, the next term
is always twice as big as the one before it.
7. The 12th term is twice as big
as which term?
8. If the current term is the 15th 9. If the current term is the 10th
term, what term came before
term, what term came before
it?
it?
10. If the current term is the 20th term, what term came before it? Write your answer as a subtraction
problem.
11. If the current term is the 𝑛th term, what term came before it? Write your answer as a subtraction
problem.
Writing a Recursive Equation. We want to write an equation that says: “The 𝑛th term is two times
bigger than the term before it.” Use problem 11 to help you write a recursive equation for our zombie
sequence.
𝑍𝑛 =
Now let’s make the Zombiepocalypse even worse. Pretend that we still start with one zombie, but
instead of only eating once per day, now our zombie eats three times per day. This means that each
zombie will turn 3 humans into zombies every day.
12. Write a new sequence, 𝐻𝑛 (for Hungry zombies). Remember to use curly braces around your
sequence. Compute at least the first ten terms.
13. Write a recursive equation to find 𝐻𝑛 in terms of 𝐻𝑛−1
14. Use your resursive formula to find 𝐻15.
15. How does 𝐻15 compare to 𝑍15 ? Which is larger? How much larger is it?
16. When does 𝐻𝑛 = 𝑍𝑛 ? Does it look like there is a pattern to when this happens? Explain your
thinking!