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Question

Is the sequence arithmetic or geometric?
Does it have a common difference or common ratio?
What is the common difference or common ratio?
What are the next three terms?
64, 32, 16, 8, 4, . . .

Answer 1

The given sequence is: \( 64, 32, 16, 8, 4, \ldots \)

1. Is the sequence arithmetic or geometric?

   • The sequence is geometric.

2. Does it have a common difference or common ratio?

   • It has a common ratio.

3. What is the common difference or common ratio?

   • The common ratio \( r \) can be calculated by dividing any term by the previous term:
     • \( \frac{32}{64} = \frac{1}{2} \)
     • \( \frac{16}{32} = \frac{1}{2} \)
     • \( \frac{8}{16} = \frac{1}{2} \)
     • \( \frac{4}{8} = \frac{1}{2} \)
   • Thus, the common ratio is \( r = \frac{1}{2} \).

4. What are the next three terms?

   • To find the next terms, we multiply the last term by the common ratio:
     • \( 4 \times \frac{1}{2} = 2 \)
     • \( 2 \times \frac{1}{2} = 1 \)
     • \( 1 \times \frac{1}{2} = \frac{1}{2} \)
   • Therefore, the next three terms are: \( 2, 1, \frac{1}{2} \).

So, the complete answer is:

• The sequence is geometric.
• It has a common ratio of \( \frac{1}{2} \).
• The next three terms are \( 2, 1, \frac{1}{2} \).