1. Definition of a Geometric Series
A geometric series is the sum of the terms of a geometric sequence. Instead of listing the numbers, we add them to find a total value. This total is often written as \( S_n \) for the first \( n \) terms.
Example:
Sequence: \( 2, 6, 18, 54 \)
Series: \( 2 + 6 + 18 + 54 = 80 \)
The pattern is based on multiplying by the same common ratio.
2. Sum of First \(n\) Terms of a GP
The sum of the first \( n \) terms of a geometric sequence depends on the common ratio \( r \). There are two main cases.
2.1. Case \(r \neq 1\)
When the common ratio is not equal to 1, the sum of the first \( n \) terms is:
\( S_n = a \left( \dfrac{1 - r^n}{1 - r} \right) \)
This formula comes from multiplying and subtracting two versions of the series to eliminate intermediate terms.
Example:
\( a = 3, r = 2, n = 4 \)
\( S_4 = 3 \left( \dfrac{1 - 2^4}{1 - 2} \right) = 3 \left( \dfrac{1 - 16}{-1} \right) = 45 \)
2.2. Case \(r = 1\)
If \( r = 1 \), all terms in the sequence are the same. So the sum is simply:
\( S_n = n \cdot a \)
Example:
\( 5, 5, 5, 5, 5 \)
Sum of 5 terms = \( 5 \times 5 = 25 \).
3. Sum of an Infinite GP
Some geometric series continue forever. In special cases, even though the terms never end, the total sum approaches a fixed value. This is called the sum of an infinite geometric series.
3.1. Condition \(|r| < 1\)
An infinite geometric series has a finite sum only if the common ratio is between -1 and 1.
That is:
\( |r| < 1 \)
When this happens, the terms become smaller and smaller, eventually approaching zero.
3.2. Formula for Infinite Sum
If \(|r| < 1\), the sum of an infinite GP is:
\( S_{\infty} = \dfrac{a}{1 - r} \)
This formula works because the terms shrink rapidly and the remaining partial sum gets closer to a fixed value.
Example:
Sequence: \( 8, 4, 2, 1, 0.5, \ldots \)
\( S_{\infty} = \dfrac{8}{1 - 1/2} = 16 \)
4. Applications of GP Sum
The formulas for geometric series help solve many types of problems, such as finding unknown terms, the number of terms, or the common ratio.
4.1. Finding Number of Terms
If the sum and first term are known, you can solve for \( n \) by substituting in the formula for \( S_n \) and solving the resulting equation.
Example:
If \( S_n = 121 \), \( a = 1 \), \( r = 3 \):
\( 121 = \dfrac{1(1 - 3^n)}{1 - 3} \)
Solving gives \( 3^n = 121 \Rightarrow n \approx 4.9 \). Since \( n \) must be a whole number, the exact sum does not correspond to a complete term count.
4.2. Finding First Term or Common Ratio
Rearranging the sum formula helps find missing values. Substitute known values and solve for the unknown.
Example:
Given \( S_4 = 60 \), \( r = 2 \):
\( 60 = a \left( \dfrac{1 - 2^4}{1 - 2} \right) \Rightarrow 60 = 15a \Rightarrow a = 4 \)
5. Examples
These examples show how geometric series can grow quickly or approach a limit depending on the common ratio.
5.1. Simple GP Sum Examples
- Sum of first 5 terms of \( 2, 6, 18, 54, \ldots \)
- Sum of first 6 terms of \( 5, 10, 20, 40, \ldots \)
- Sum of first 4 terms of \( 81, 27, 9, 3, \ldots \)
Each uses either finite-sum formulas or simple ratio patterns.
5.2. Real-Life Applications of GP Series
Geometric series appear naturally in many real-world patterns:
- Repeated percentage growth (interest, population)
- Light intensity fading by a constant factor
- Successive reflections or echoes reducing in strength
These patterns grow or decay based on multiplication, not addition.