Answer:
- geometric
- common ratio: -5
- a[n]=-5a[n-1]; 625
Step-by-step explanation:
<h3>A.</h3>
We start this problem the way we start all problems: we read the problem statement and observe the given data and relationships.
Here, we observe that the given numbers are all multiples (powers) of 5, <em>with alternating signs</em>. We know that an arithmetic sequence cannot have alternating signs, so this is <em>not an arithmetic sequence</em>.
We are familiar with the first few powers of 5, so we recognize the terms have a common ratio of -5. If terms have a common ratio, they are a geometric sequence, so this is a geometric sequence.
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<h3>B.</h3>
As we observed in Part A, the common ratio is -5.
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<h3>C.</h3>
The full recursive formula will include the initial value in the sequence:
a[1] = -5
a[n] = -5·a[n-1]
Then the next value is ...
a[4] = -5·a[3] = -5·(-125)
a[4] = 625
