Program to Check Geometric Progression

A sequence of numbers is called a Geometric progression if the ratio of any two consecutive terms is always the same.

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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:

Where,
            a = First term
            r = common ratio
            ar^n-1 = nth term

Example: 

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:

1. Finite geometric progression
2. 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. 

Example:

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.

Example:

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:

1. Initial term: In a geometric progression, the first number is called the initial term.
2. Common ratio: The ratio between a term in the sequence and the term before it is called the "common ratio." 
3. 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 nthterm 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. 

Approach: 

1. First calculate the common ratio r by arr[1] / arr[0]
2. Iterate over an array and calculate the ratio of the consecutive terms.
3. Check if the calculated ratio is not equal to the common ratio r
   • Return false
4. 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:

True

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

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