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:
GP-series
Where,
a = First term
r = common ratio
arn-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:
- 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.
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:
- 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:
Nth term of a Geometric Progression
The formula for the sum of the nth term of Geometric Progression:
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:
- First calculate the common ratio r by arr[1] / arr[0]
- 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
- After traversal, if the calculated ratio is equal to the common ratio r every time
Below is the implementation of the above approach:
C++
#include <bits/stdc++.h>
using namespace std;
bool is_geometric(int arr[], int n)
{
if (n == 1)
return true;
int ratio = arr[1] / (arr[0]);
for (int i = 1; i < n; i++) {
if ((arr[i] / (arr[i - 1])) != ratio) {
return false;
}
}
return true;
}
int main()
{
int arr[] = { 2, 6, 18, 54 };
int n = sizeof(arr) / sizeof(arr[0]);
(is_geometric(arr, n)) ? (cout << "True" << endl)
: (cout << "False" << endl);
return 0;
}
|
Java
import java.util.Arrays;
class GFG {
static boolean is_geometric(int arr[], int n)
{
if (n == 1)
return true;
int ratio = arr[1] / (arr[0]);
for (int i = 1; i < n; i++) {
if ((arr[i] / (arr[i - 1])) != ratio) {
return false;
}
}
return true;
}
public static void main(String[] args)
{
int arr[] = { 2, 6, 18, 54 };
int n = arr.length;
if (is_geometric(arr, n))
System.out.println("True");
else
System.out.println("False");
}
}
|
Python3
def is_geometric(li):
if len(li) <= 1:
return True
ratio = li[1]/float(li[0])
for i in range(1, len(li)):
if li[i]/float(li[i-1]) != ratio:
return False
return True
print(is_geometric([2, 6, 18, 54]))
|
C#
using System;
class Geeks {
static bool is_geometric(int[] arr, int n)
{
if (n == 1)
return true;
int ratio = arr[1] / (arr[0]);
for (int i = 1; i < n; i++) {
if ((arr[i] / (arr[i - 1])) != ratio) {
return false;
}
}
return true;
}
public static void Main(String[] args)
{
int[] arr = new int[] { 2, 6, 18, 54 };
int n = arr.Length;
if (is_geometric(arr, n))
Console.WriteLine("True");
else
Console.WriteLine("False");
}
}
|
PHP
<?php
function is_geometric($arr)
{
if (sizeof($arr) <= 1)
return True;
# Calculate ratio
$ratio = $arr[1]/$arr[0];
# Check the ratio of the remaining
for($i=1; $i<sizeof($arr); $i++)
{
if (($arr[$i]/($arr[$i-1])) != $ratio)
{
return "Not a geometric sequence";
}
}
return "Geometric sequence";
}
$my_arr1 = array(2, 6, 18, 54);
print_r(is_geometric($my_arr1)."\n");
print_r(is_geometric($my_arr2)."\n");
?>
|
Javascript
<script>
function is_geometric(arr, n)
{
if (n == 1)
return true;
let ratio = parseInt(arr[1] / (arr[0]));
for(let i = 1; i < n; i++)
{
if (parseInt((arr[i] /
(arr[i - 1]))) != ratio)
{
return false;
}
}
return true;
}
let arr = [ 2, 6, 18, 54 ];
let n = arr.length;
(is_geometric(arr, n)) ?
(document.write("True")) :
(document.write("False"));
</script>
|
Time Complexity: O(n), Where n is the length of the given array.
Auxiliary Space: O(1)
Basic Program related to Geometric Progression
More problems related to Geometric Progression
Recent Articles on Geometric Progression!
Feeling lost in the world of random DSA topics, wasting time without progress? It's time for a change! Join our DSA course, where we'll guide you on an exciting journey to master DSA efficiently and on schedule.
Ready to dive in? Explore our Free Demo Content and join our DSA course, trusted by over 100,000 geeks!