A geometric series is the sum of the terms in a geometric sequence. If the sequence has a definite number of terms, the simple formula for the sum is
Formula 3: 
This form of the formula is used when the number of terms ( n), the first term ( a 1), and the common ratio ( r) are known.
Another formula for the sum of a geometric sequence is
Formula 4: 
This form requires the first term ( a 1), the last term ( a n ), and the common ratio ( r) but does not require the number of terms ( n).
Example 1
Find the sum of the first five terms of the geometric sequence in which a 1 = 3 and r = –2.
a 1 = 3, r = –2, n = 5
Use Formula 4: 
Example 2
Find the sum of the geometric sequence for which 
.
Use Formula 4: 
Example 3
Find a 1 in each geometric series described.
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S n = 244, r = –3, n = 5
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S n = 15.75, r = 0.5, a n = 0.25
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S n = 244, r = –3, n = 5
Use Formula 3: 
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S n = 15.75, r = 0.5, a n = 0.25
Use Formula 4: 
Formula 5:
If a geometric series is infinite (that is, endless) and –1 < r < 1, then the formula for its sum becomes 
If r > 1 or if r < –1, then the infinite series does not have a sum.
Example 4
Find the sum of each of the following geometric series.
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25 + 20 + 16 + 12.8 + …
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3 – 9 + 27 – 81 + …
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25 + 20 + 16 + 12.8 + …
First find r.
Since 
, this infinite geometric series has a sum.
Use Formula 5.
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3 – 9 + 27 – 81 + …
First find r.
Since –3 < –1, this geometric series does not have a sum.