A sequence of numbers is called a Geometric progression if the ratio of any two consecutive terms is always the same.
In simple terms, A geometric series is a list of numbers where each number, or term, is found by multiplying the previous term by a common ratio r. The general form of Geometric Progression is:
a = First term
r = common ratio
arn-1 = nth term
The sequence 2, 4, 8, 16 is a GP because ratio of any two consecutive terms in the series (common difference) is same (4 / 2 = 8 / 4 = 16 / 8 = 2).
The geometric progression is of two types:
- Finite geometric progression
- Infinite geometric progression.
1. Finite geometric progression
In finite geometric progression contains a finite number of terms. The last term is always defined in this type of progression.
The sequence 1/2,1/4,1/8,1/16,…,1/32768 is a finite geometric series where the first term is 1/2 and the last term is 1/32768.
2. Infinite geometric progression
Infinite geometric progression contains an infinite number of terms. The last term is not defined in this type of progression.
Sequence 3, 9, 27, 81, … is an infinite series where the first term is 3 but the last term is not defined.
Fact about Geometric Progression:
- Initial term: In a geometric progression, the first number is called the initial term.
- Common ratio: The ratio between a term in the sequence and the term before it is called the “common ratio.”
- The behavior of a geometric sequence depends on the value of the common ratio. If the common ratio is:
- Positive, the terms will all be the same sign as the initial term.
- Negative, the terms will alternate between positive and negative.
- Greater than 1, there will be exponential growth towards positive or negative infinity (depending on the sign of the initial term).
- 1, the progression is a constant sequence.
- Between -1 and 1 but not zero, there will be exponential decay towards zero.
- -1, the progression is an alternating sequence.
- Less than -1, for the absolute values there is exponential growth towards (unsigned) infinity, due to the alternating sign.
The formula for the nth term of a Geometric Progression:
If ‘a1′ is the first term and ‘r’ is the common ratio. Thus, the explicit formula for nth term of finite GP series:
The formula for the sum of the nth term of Geometric Progression:
How do we check whether a series is a Geometric progression or not?
The property of the GP series is that the ratio of the consecutive terms is same.
- First calculate the common ratio r by arr / arr
- Iterate over an array and calculate the ratio of the consecutive terms.
- Check if the calculated ratio is not equal to the common ratio r
- Return false
- After traversal, if the calculated ratio is equal to the common ratio r every time
- Return true
Below is the implementation of the above approach:
Time Complexity: O(n), Where n is the length of the given array.
Auxiliary Space: O(1)
Basic Program related to Geometric Progression
- Program to print GP (Geometric Progression)
- Program for sum of geometric series
- Find the missing number in Geometric Progression
- Program for N-th term of Geometric Progression series
- Find all triplets in a sorted array that forms Geometric Progression
- Removing a number from array to make it Geometric Progression
- Minimum number of operations to convert a given sequence into a Geometric Progression
- Number of GP (Geometric Progression) subsequences of size 3
More problems related to Geometric Progression
- Find the sum of series 3, -6, 12, -24 . . . upto N terms
- Find the sum of the series 2, 5, 13, 35, 97
- Area of squares formed by joining mid points repeatedly
- Longest Geometric Progression