# Lesson 10

Situations and Sequence Types

### 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

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

- Complete the table with the area of the piece of paper \(A(n)\), in square inches, after it is folded in half \(n\) times.
- Define \(A\) for the \(n^{\text{th}}\) term.
- 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:

- Describe how the number of dots increases from Stage 1 to Stage 3.
- Write a definition for sequence \(D\), so that \(D(n)\) is the number of dots in Stage \(n\).
- 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.

- 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.
- 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.)