The formula for the sum of a geometric sequence depends on whether the sequence is finite or infinite. According to the provided reference, there are two main formulas to consider.
Geometric Sum Formulas
Here's a breakdown of the formulas:
Finite Geometric Series
A finite geometric series has a specific number of terms. The formula to calculate its sum, Sn, is:
- If r = 1: Sn = an*
- If r ≠ 1: Sn = a(1 - rn) / (1 - r)
Where:
- Sn is the sum of the first n terms.
- a is the first term of the sequence.
- r is the common ratio (the value you multiply each term by to get the next term).
- n is the number of terms in the sequence.
Example: Find the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54, 162.
- a = 2
- r = 3
- n = 5
Therefore, S5 = 2(1 - 35) / (1 - 3) = 2(1 - 243) / (-2) = 2(-242) / (-2) = 242
Infinite Geometric Series
An infinite geometric series continues indefinitely. The formula for the sum of an infinite geometric series only converges (approaches a finite value) if the absolute value of the common ratio, |r|, is less than 1 (i.e., -1 < r < 1). The formula is:
- Sn = a / (1 - r)
Where:
- Sn is the sum of the infinite series.
- a is the first term of the sequence.
- r is the common ratio.
Example: Find the sum of the infinite geometric series 4 + 2 + 1 + 1/2 + ...
- a = 4
- r = 1/2
Therefore, Sn = 4 / (1 - 1/2) = 4 / (1/2) = 8
