The geometric partial sum is found using a straightforward formula that requires only the first term and the common ratio of the geometric sequence.
Understanding Geometric Sequences and Partial Sums
A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, 2, 4, 8, 16... is a geometric sequence with a common ratio of 2.
A partial sum is the sum of a finite number of terms from a sequence. The nth partial sum (denoted Sn) is the sum of the first n terms.
The Formula for the Geometric Partial Sum
The formula for calculating the nth partial sum of a geometric sequence is:
Sn = a1(1 - rn) / (1 - r)
Where:
- Sn is the nth partial sum
- a1 is the first term of the sequence
- r is the common ratio
- n is the number of terms
This formula is derived from the sum of a geometric series. It works for any geometric sequence where the common ratio (r) is not equal to 1.
Examples
Let's look at a couple of examples:
Example 1: Find the sum of the first 5 terms of the geometric sequence 3, 6, 12, 24...
- a1 = 3
- r = 2
- n = 5
Substitute these values into the formula:
S5 = 3(1 - 25) / (1 - 2) = 3(1 - 32) / (-1) = 3(-31) / (-1) = 93
Therefore, the sum of the first 5 terms is 93.
Example 2: Find the sum of the first 4 terms of the geometric sequence 100, 20, 4, 0.8...
- a1 = 100
- r = 0.2
- n = 4
Substitute into the formula:
S4 = 100(1 - 0.24) / (1 - 0.2) = 100(1 - 0.0016) / 0.8 = 100(0.9984) / 0.8 = 124.8
The sum of the first 4 terms is 124.8.
Practical Applications
The formula for the geometric partial sum has many applications in various fields, including:
- Finance: Calculating compound interest, annuities, and loan repayments.
- Physics: Modeling exponential growth and decay.
- Computer Science: Analyzing algorithms and data structures.
