To find out if a sequence is arithmetic when given 4 terms:

1. Subtract 2nd term minus 1st term.
2. Subtract 3rd term minus 2nd term.
3. Subtract 4th term minus 3rd term.

If all three are the same then the sequence is arithmetic,
with d = common difference = this value.

To find out if a sequence is geometric when given 4 terms:
1. Divide 2nd term by 1st term.
2. Divide 3rd term by 2nd term.
3. Divide 4th term by 3rd term.

If all three are the same then the sequence is geometric,
with r = common ratio = this value.

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First we test to see if it's an arithmetic sequence:

1. Subtract 2nd term minus 1st term.  (5/9) - (-5/3) = 20/9  
2. Subtract 3rd term minus 2nd term. (-5/27) - (5/9) = -20/27 
3. No need to subtract 4th term minus 3rd term since those aren't the
  same.  We know it is not an arithmetic sequence.

So that rules out choices 2 and 4.  So it's between 1 and 3.

Next we test to see if it's a geometric sequence:

1. Divide 2nd term by 1st term.  (5/9)÷(-5/3) = (5/9)(-3/5) = -1/3
2. Divide 3rd term by 2nd term.  (-5/27)÷(5/9)= (-5/27)(9/5) = -1/3
3. Divide 4th term by 3rd term.  (5/81)÷(-5/27) = (5/81)(-27/5) = -1/3 

All three are the same, so the sequence is geometric,
with r = -1/3.

The correct choice is 1.

Edwin