A sequence of numbers is called a Geometric progression if the ratio of any two consecutive terms is always same. In simple terms, it means that next number in the series is calculated by multiplying a fixed number to the previous number in the series.For example, 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).
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 behaviour 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.
Formula of nth term of an Geometric Progression :
If ‘a’ is the first term and ‘r’ is the common ratio.Thus, the explicit formula is
Formula of sum of nth term of Geometric Progression:
How we check whether a series is geometric progression or not?
Let’s find the ratio of the consecutive terms, so we can say that the ratio of the consecutive terms of given sequences is 13 or a constant. So this sequence is forming a geometrical progression.
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