The sum of a geometric progression is given by the formula S = a(1 – rn) / (1 – r), or equivalently S = a(rn – 1) / (r – 1).
Understanding the Formula
This formula is used to calculate the sum of the first n terms of a geometric series, where:
- S is the sum of the series.
- a is the first term of the series.
- r is the common ratio between consecutive terms.
- n is the number of terms being summed.
Derivation Overview
According to the reference, the formula arises from understanding the relationship of terms within an arithmetico geometric series, culminating in the given formula S = a(1 – rn) / (1 – r), or S = a(rn – 1) / (r – 1) . Note that the formula presented in the reference is specifically for the sum of n terms, and not the sum of infinite geometric series.
Key Components:
- First Term (a): The starting value of the sequence. For example, in the sequence 2, 4, 8, 16,... '2' would be 'a'.
- Common Ratio (r): The constant factor by which each term is multiplied to get the next term. In the example 2, 4, 8, 16, the common ratio 'r' is '2'.
- Number of Terms (n): How many terms are being added together. If you are summing the first 4 terms of the example sequence 2,4,8,16, then 'n' would be 4.
Example Application:
Let's consider a simple geometric series: 3 + 6 + 12 + 24. Here, a=3, r=2, and n=4.
Using the formula S = a(rn – 1) / (r – 1):
S = 3(24 - 1) / (2-1)
S = 3(16-1)/1
S=3 * 15 = 45
Hence, the sum of the first four terms is 45, which you can verify manually (3+6+12+24=45).
Practical Insights
- The formula is invaluable for quickly calculating the sum of a large number of terms without needing to add each term individually.
- Be mindful when 'r' is equal to 1, the formula cannot be used because the denominator will be zero. When r =1, the sequence becomes an arithmetic progression with a common difference of 0 and the sum of ‘n’ terms is simply n*a.
- This formula is fundamental in various mathematical and financial calculations such as compound interest and annuity calculations.
| Term | Description |
|---|---|
| S | Sum of the series |
| a | First term of the series |
| r | Common ratio |
| n | Number of terms |
