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Geometric Sequences
We don’t really think about it, but we rely on the basketball’s ability to bounce back to a specific height.
If basketballs bounced to radically different heights, we couldn’t do things like this:
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Kaleb’s Work
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Sandra’sWork
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Try one
Consider the following sequence:
4, 8, 16, 32, …
a. Write a recursive rule for this sequence.
b. Write an explicit rule for this sequence.
c. Use one of your rules to find the 6th term in the sequence.
a.Let n = number of term
Let f(n) = value of term n
f(1) = 10
f(n) = 3 ∙ f(n) for integers n ≥ 2
b.f(n+1) = 10 ∙ 3(n) for integers n ≥ 1
c. f(6) = 10 ∙ 3(6)
f(6) = 10 ∙ 729
f(6) = 7290
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Geometric Sequences
Your cell phone is covered in bacteria. The same bacteria from your fingers
and you face also live on your cell phone. A culture of bacteria swabbed from
a cell phone has 150 of a particular bacteria on it. That particular bacteria
doubles every day.
a. Write a recursive rule to represent this situation.
b. Write an explicit rule to represent this situation.
c. Use both of your rules to determine the total amount of bacteria after
8 days.
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Geometric Sequences
Geometric Sequence: is a sequence of numbers such that each term is given by a common
multiple, r, of the previous term. f(n + 1) = r · f(n) where n is the number of the term.
Recursive rule for a geometric sequence:
Let n be the number of the term. Let f(n) be the value of term n. Let p be the value of the
first term
f(1) = p
f(n + 1) = r · f(n) for integers n ≥ 2
Explicit rule for a geometric sequence:
f(n) = (r)n-1 · f(1), for integers n ≥ 1
or
f(n + 1) = (r)n · f(1), for integers n ≥ 1
or a closed formula:
f(n) = f(0) · rn, for integers n ≥ 1
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Recall.. Arithmetic Sequences
• Arithmetic Sequence: a sequence of numbers such that each term is given by a common
difference, d, of the previous term. f(n + 1) = f(n) + d where n is the number of the term.
• Recursive rule for an arithmetic sequence: .
Let n be the number of the term. Let f(n) be the value of term n. Let p be the value of the first term
f(1) = p
f(n + 1) = f(n) + d for integers n ≥ 2
or
f(1) = p
f(n) = f(n - 1) + d for integers n ≥ 2
• Explicit Rule for an arithmetic sequence:
f(n + 1) = d(n) + f(1), for integers n ≥ 1
or
f(n) = d(n-1) + f(1), for integers n ≥ 1
• or a closed formula:
• f(n) = dn + f(0), for integers n ≥ 1
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In your groups, take turns to matching a situation card to one of the function cards.
For each recursive function, define n and f(n).
For each blank card, fill in the appropriate information to create a matched set.
If you place a card, explain why that situation matches the function.
Everyone in your group should agree on, and be able to explain, your choice.
Then, take turns matching your explicit function to your matched pair of recursive
functions and situations.
• Wherever there are blank cards, create an explicit or recursive function to make a
matching set.
• When everyone agrees, glue the cards across as a single row in the respective
columns on your paper.
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