Lesson 7

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.

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.

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.)

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.)

Match each recursive definition with one of the sequences.

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?