Geometric Progression (GP)
Geometric Progression (GP)
A Geometric Progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This sequence is used in various areas of mathematics, including finance, physics, and other sciences.
Definition
A GP is defined by its first term ( a ) and the common ratio ( r ). The sequence is:
[ a, ar, ar^2, ar^3, \ldots ]
General Form
The ( n )-th term of a GP can be expressed as:
[ T_n = ar^{n-1} ]
where:
- ( T_n ) is the ( n )-th term,
- ( a ) is the first term,
- ( r ) is the common ratio,
- ( n ) is the term number.
Sum of a GP
The sum of the first ( n ) terms of a GP (denoted as ( S_n )) can be calculated using the formula:
[ S_n = \frac{a(1 - r^n)}{1 - r} \quad \text{for } r \neq 1 ]
If ( r = 1 ), the sum of the first ( n ) terms is simply:
[ S_n = na ]
Infinite GP
For an infinite GP, the sum converges to a finite value if the common ratio ( r ) has an absolute value less than 1. The sum of an infinite GP is given by:
[ S_{\infty} = \frac{a}{1 - r} \quad \text{for } |r| < 1 ]
Differences and Important Points
| Aspect | Description |
|---|---|
| First Term | Denoted by ( a ), it is the starting point of the progression. |
| Common Ratio | Denoted by ( r ), it is the factor by which we multiply to get the next term. |
| ( n )-th Term | Given by ( ar^{n-1} ), it represents the term at position ( n ). |
| Sum of First ( n ) Terms | Given by ( \frac{a(1 - r^n)}{1 - r} ) for ( r \neq 1 ), it is the sum of the series up to ( n ) terms. |
| Infinite GP | If ( |
| Divergence | If ( |
Examples
Example 1: Finding the ( n )-th Term
Given a GP with the first term ( a = 3 ) and common ratio ( r = 2 ), find the 5th term.
[ T_5 = ar^{5-1} = 3 \cdot 2^{4} = 3 \cdot 16 = 48 ]
Example 2: Sum of First ( n ) Terms
Find the sum of the first 4 terms of the GP from Example 1.
[ S_4 = \frac{3(1 - 2^4)}{1 - 2} = \frac{3(1 - 16)}{-1} = \frac{-45}{-1} = 45 ]
Example 3: Sum of an Infinite GP
Consider an infinite GP with ( a = 5 ) and ( r = \frac{1}{2} ). Find the sum.
[ S_{\infty} = \frac{5}{1 - \frac{1}{2}} = \frac{5}{\frac{1}{2}} = 5 \cdot 2 = 10 ]
Example 4: Divergence of an Infinite GP
If we have an infinite GP with ( a = 1 ) and ( r = 2 ), the sum diverges because ( |r| \geq 1 ).
In conclusion, understanding the properties and formulas of Geometric Progression is crucial for solving problems related to this topic. By recognizing the pattern of multiplication by a common ratio, one can determine any term in the sequence, as well as the sum of a finite or infinite number of terms, provided the conditions for convergence are met.