How do I define a recursive rule for the geometric sequence 10, 80, 640, 5120?

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To define a recursive rule for the geometric sequence 10, 80, 640, 5120, we must first recognize the pattern of the sequence.

In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor, which is called the common ratio. In this case, we can determine the common ratio by dividing each term by the one before it:

  • 80 / 10 = 8
  • 640 / 80 = 8
  • 5120 / 640 = 8

We see that the common ratio (r) is consistently 8.

Now, we can formulate the recursive rule as follows:

  1. The first term of the sequence is given by: a1 = 10.
  2. Each subsequent term can be found using the previous term: an = an-1 * 8 for n > 1.

This recursive rule effectively captures the relationship between the terms in the geometric sequence. To summarize, the rule can be stated simply as:

a1 = 10
an = an-1 * 8 for n > 1

With these rules, one can easily calculate the terms of the sequence:

  • First term: 10
  • Second term: 10 * 8 = 80
  • Third term: 80 * 8 = 640
  • Fourth term: 640 * 8 = 5120

These calculations confirm that the recursive rule accurately describes the sequence. Following this approach, you can create similar rules for other geometric sequences by identifying their first term and common ratio.

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