# Lesson 11

Adding Up

- Let’s look at sequences and the sum of their terms.

### Problem 1

Find the sum of the sequence:

- \(\frac 1 3 + \frac 1 9\)
- \(\frac 2 3 + \frac 2 9\)
- \(\frac 1 3 + \frac 1 9 + \frac{1}{27}\)
- \(\frac 2 3 + \frac 2 9 + \frac{2}{27}\)
- \(\frac 1 3 + \frac 1 9 + \frac{1}{27} + \frac{1}{81}\)
- \(\frac 2 3 + \frac 2 9 + \frac{2}{27} + \frac{2}{81}\)

### Problem 2

Priya is walking down a long hallway. She walks halfway and stops. Then, she walks half of the remaining distance, and stops again. She continues to stop every time she goes half of the remaining distance.

- What fraction of the length of the hallway will Priya have covered after she starts and stops two times?
- What fraction of the length of the hallway will Priya have covered after she starts and stops four times?
- Will Priya ever reach the end of the hallway, repeatedly starting and stopping at half the remaining distance? Explain your thinking.

### Problem 3

A geometric sequence \(h\) starts with 10, 5, . . . Explain how you would calculate the value of the 100th term.

### Problem 4

Here is a graph of sequence \(r\). Define \(r\) recursively using function notation.

### Problem 5

An unfolded piece of paper is 0.05 mm thick.

- Complete the table with the thickness of the piece of paper \(T(n)\) after it is folded in half \(n\) times.
- Define \(T\) for the \(n^{\text{th}}\) term.
- What is a reasonable domain for the function \(T\)? Explain how you know.

\(n\) | \(T(n)\) |
---|---|

0 | 0.05 |

1 | |

2 | |

3 |

### Problem 6

A piece of paper is 0.05 mm thick.

- Complete the table with the thickness of the paper \(t(n)\), in mm, after it has been folded \(n\) times.
- Does \(t(0.5)\) make sense? Explain how you know.

\(n\) | \(t(n)\) |
---|---|

0 | \(0.05\) |

1 | |

2 | |

3 |

### Problem 7

An arithmetic sequence \(a\) starts 84, 77, . . .

- Define \(a\) recursively.
- Define \(a\) for the \(n^{\text{th}}\) term.

### Problem 8

Here is a pattern of growing rectangles:

- Describe how the rectangle grows from Step 0 to Step 2.
- Write an equation for sequence \(S\), so that \(S(n)\) is the number of squares in Step \(n\).
- Is \(S\) a geometric sequence, an arithmetic sequence, or neither? Explain how you know.