Lesson 7

Representing More Sequences

  • Let's learn about Info Gaps

Problem 1

Here is the recursive definition of a sequence: \(f(1) = 10, f(n) = f(n-1) - 1.5\) for \(n\ge2\).

  1. Is this sequence arithmetic, geometric, or neither?
  2. List at least the first five terms of the sequence.
  3. Graph the value of the term \(f(n)\) as a function of the term number \(n\) for at least the first five terms of the sequence.

Problem 2

An arithmetic sequence \(k\) starts 12, 6, . . .

  1. Write a recursive definition for this sequence.
  2. Graph at least the first five terms of the sequence.

Problem 3

An arithmetic sequence \(a\) begins 11, 7, . . .

  1. Write a recursive definition for this sequence using function notation.
  2. Sketch a graph of the first 5 terms of \(a\).
  3. Explain how to use the recursive definition to find \(a(100)\). (Don't actually determine the value.)
(From Unit 1, Lesson 6.)

Problem 4

A geometric sequence \(g\) starts 80, 40, . . .

  1. Write a recursive definition for this sequence using function notation.
  2. Use your definition to make a table of values for \(g(n)\) for the first 6 terms.
  3. Explain how to use the recursive definition to find \(g(100)\). (Don't actually determine the value.)
(From Unit 1, Lesson 6.)

Problem 5

Match each recursive definition with one of the sequences.

A:

\(h(1)=1, h(n)=2 \boldcdot h(n-1) + 1\) for \(n\ge2\)

B:

\(p(1) = 1, p(n) = 2 \boldcdot p(n-1)\) for \(n\ge2\)

C:

\(a(1) = 80, a(n) = \frac 1 2 \boldcdot a(n-1)\) for \(n\ge2\)

1:

80, 40, 20, 10, 5

2:

1, 2, 4, 8, 16

3:

1, 3, 7, 15, 31

(From Unit 1, Lesson 5.)

Problem 6

For each sequence, decide whether it could be arithmetic, geometric, or neither. 

  1. 25, 5, 1, . . .
  2. 25, 19, 13, . . .
  3. 4, 9, 16, . . .
  4. 50, 60, 70, . . .
  5. \(\frac{1}{2},\) 3, 18, . . .

For each sequence that is neither arithmetic nor geometric, how can you change a single number to make it an arithmetic sequence? A geometric sequence?

(From Unit 1, Lesson 3.)