Lesson 10

Situations and Sequence Types

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Problem 1

A sequence is defined by \(f(0) = 3, f(n) = 2 \boldcdot f(n-1)\) for \(n\ge1\). Write a definition for the \(n^{\text{th}}\) term of \(f\).

Solution

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Problem 2

A geometric sequence, \(g(n)\) starts 20, 60, . . . Define \(g\) recursively and for the \(n^{\text{th}}\) term.

Solution

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Problem 3

A geometric sequence \(g\) starts at 500 and has a growth factor of 0.6. Sketch a graph of \(g\) showing the first 5 terms.

 

 

Solution

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(From Unit 1, Lesson 7.)

Problem 4

  1. An arithmetic sequence has \(a(1)=4\) and \(a(2)=16\). Explain or show how to find the value of \(a(15)\)
  2. A geometric sequence has \(g(0)=4\) and \(g(1)=16\). Explain or show how to find the value of \(g(15)\).

Solution

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(From Unit 1, Lesson 8.)

Problem 5

A piece of paper has an area of 96 square inches.

  1. Complete the table with the area of the piece of paper \(A(n)\), in square inches, after it is folded in half \(n\) times.
  2. Define \(A\) for the \(n^{\text{th}}\) term.
  3. What is a reasonable domain for the function \(A\)? Explain how you know.
\(n\) \(A(n)\)
0 96
1  
2  
3  

Solution

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(From Unit 1, Lesson 9.)

Problem 6

Here is a growing pattern:

A growing pattern with three stages.
  1. Describe how the number of dots increases from Stage 1 to Stage 3.
  2. Write a definition for sequence \(D\), so that \(D(n)\) is the number of dots in Stage \(n\).
  3. Is \(D\) a geometric sequence, an arithmetic sequence, or neither? Explain how you know.

Solution

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(From Unit 1, Lesson 9.)

Problem 7

A paper clip weighs 0.5 grams and an empty envelope weighs 6.75 grams.

  1. Han adds paper clips one at a time to an empty envelope. Complete the table with the weight of the envelope \(w(n)\), in grams, after \(n\) paper clips have been added.
  2. Does \(w(10.25)\) make sense? Explain how you know.
\(n\) \(w(n)\)
0 \(6.75\)
1  
2  
3  

Solution

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(From Unit 1, Lesson 9.)