A recursive rule for a geometric sequence is used to express each term based on the previous term. In the case of the sequence 10, 80, 640, 5120, we start by identifying the common ratio between the terms.
To find the common ratio, we divide the second term by the first term:
Common ratio = 80 / 10 = 8
Next, we can verify this ratio by checking the other consecutive terms:
- 640 / 80 = 8
- 5120 / 640 = 8
Since the ratio is consistent throughout the sequence, we can express the recursive rule as follows:
a1 = 10
an = an-1 × 8 (for n > 1)
In this rule, a1 is the first term of the sequence, which is 10, and each subsequent term an is found by multiplying the previous term an-1 by 8. This reflects the definition of a geometric sequence, where each term is multiplied by a constant ratio to get the next term.