Arithmetic Progression - GeeksforGeeks (2024)

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A sequence of numbers is called an Arithmetic progression if the difference between any two consecutive terms is always the same.

In simple terms, it means that the next number in the series is calculated by adding a fixed number to the previous number in the series.

Example:

Sequence 1, 4, 7,10 is an AP because the difference between any two consecutive terms in the series (common difference) is the same.
The difference between 2nd and 1st term 4 – 1 = 3
The difference between 3rd and 2nd term 7 – 4 = 3
The difference between 4th and 3rd term10 – 7 = 3
From the above example, we can conclude that there is some common difference between any two consecutive elements.

Notation in Arithmetic Progression

In AP, there are some main terms that are generally used, which are denoted as:

Initial term (a): In an arithmetic progression, the first number in the series is called the initial term.
Common difference (d): The value by which consecutive terms increase or decrease is called the common difference. The behavior of the arithmetic progression depends on the common difference d. If the common difference is: positive, then the members (terms) will grow towards positive infinity or negative, then the members (terms) will grow towards negative infinity.
nth Term (a_n): The nth term of the AP series
Sum of the first n terms (S_n): The sum of the first n terms of the AP series.

The formula for the n^th term of an A.P:

If ‘a’ is the initial term and ‘d’ is a common difference. Thus, the explicit formula is

The formula for the sum of n terms of AP:

How do we check whether a series is an arithmetic progression or not?

1. Brute approach.

The idea is to sort the given array or series. After sorting, check if the differences between consecutive elements are the same or not. If all differences are the same, Arithmetic Progression is possible.

Below is the implementation of this approach:

C++

// C++ program to check if a given array// can form arithmetic progression#include <bits/stdc++.h>using namespace std;// Returns true if a permutation of arr[0..n-1]// can form arithmetic progressionbool checkIsAP(int arr[], int n){ if (n == 1) return true; // Sort array sort(arr, arr + n); // After sorting, difference between // consecutive elements must be same. int d = arr[1] - arr[0]; for (int i = 2; i < n; i++) if (arr[i] - arr[i - 1] != d) return false; return true;}// Driven Programint main(){ int arr[] = { 20, 15, 5, 0, 10 }; int n = sizeof(arr) / sizeof(arr[0]); (checkIsAP(arr, n)) ? (cout << "Yes" << endl) : (cout << "No" << endl); return 0;}

Java

// Java program to check if a given array// can form arithmetic progressionimport java.util.Arrays;class GFG { // Returns true if a permutation of // arr[0..n-1] can form arithmetic // progression static boolean checkIsAP(int arr[], int n) { if (n == 1) return true; // Sort array Arrays.sort(arr); // After sorting, difference between // consecutive elements must be same. int d = arr[1] - arr[0]; for (int i = 2; i < n; i++) if (arr[i] - arr[i - 1] != d) return false; return true; } // driver code public static void main(String[] args) { int arr[] = { 20, 15, 5, 0, 10 }; int n = arr.length; if (checkIsAP(arr, n)) System.out.println("Yes"); else System.out.println("No"); }}// This code is contributed by Anant Agarwal.

Python3

# Python3 program to check if a given # array can form arithmetic progression# Returns true if a permutation of arr[0..n-1]# can form arithmetic progressiondef checkIsAP(arr, n): if (n == 1): return True # Sort array arr.sort() # After sorting, difference between # consecutive elements must be same. d = arr[1] - arr[0] for i in range(2, n): if (arr[i] - arr[i-1] != d): return False return True# Driver codearr = [ 20, 15, 5, 0, 10 ]n = len(arr)print("Yes") if(checkIsAP(arr, n)) else print("No")# This code is contributed by Anant Agarwal.

// C# program to check if a given array// can form arithmetic progressionusing System;class GFG { // Returns true if a permutation of // arr[0..n-1] can form arithmetic // progression static bool checkIsAP(int[] arr, int n) { if (n == 1) return true; // Sort array Array.Sort(arr); // After sorting, difference between // consecutive elements must be same. int d = arr[1] - arr[0]; for (int i = 2; i < n; i++) if (arr[i] - arr[i - 1] != d) return false; return true; } // Driver Code public static void Main() { int[] arr = { 20, 15, 5, 0, 10 }; int n = arr.Length; if (checkIsAP(arr, n)) Console.WriteLine("Yes"); else Console.WriteLine("No"); }}// This code is contributed by vt_m.

JavaScript

<script>// Javascript program to check if a given array// can form arithmetic progression// Returns true if a permutation of arr[0..n-1]// can form arithmetic progressionfunction compare(a, b) { if (a < b) { return -1; } else if (a > b) { return 1; } else { return 0; }}function checkIsAP( arr, n){ if (n == 1) return true; // Sort array arr.sort(compare); // After sorting, difference between // consecutive elements must be same. let d = arr[1] - arr[0]; for (let i = 2; i < n; i++) if (arr[i] - arr[i - 1] != d) return false; return true;}// Driven Programlet arr = [ 20, 15, 5, 0, 10 ];let n = arr.length;(checkIsAP(arr, n)) ? document.write("Yes <br>") : document.write("No <br>");</script>

PHP

<?php// PHP program to check if // a given array can form// arithmetic progression// Returns true if a permutation // of arr[0..n-1] can form // arithmetic progressionfunction checkIsAP($arr, $n){ if ($n == 1) return true; // Sort array sort($arr); // After sorting, difference // between consecutive elements // must be same. $d = $arr[1] - $arr[0]; for ($i = 2; $i < $n; $i++) if ($arr[$i] - $arr[$i - 1] != $d) return false; return true;}// Driver Code$arr = array(20, 15, 5, 0, 10);$n = count($arr);if(checkIsAP($arr, $n))echo "Yes"; elseecho "No"; // This code is contributed // by Sam007?>

Output

Yes

Time Complexity: O(n Log n).
Auxiliary Space: O(1)

2. Efficient solutions

Set 1(Use Hashing)
Set 2(Use Counting Sort)

Basic Program related to Arithmetic Progression

Program for sum of arithmetic series
Program to print Arithmetic Progression series
Longest arithmetic progression with the given common difference
Check whether Arithmetic Progression can be formed from the given array
Find the missing number in Arithmetic Progression
Find N Arithmetic Means between A and B
Sum of the numbers upto N that are divisible by 2 or 5
Find First element in AP which is multiple of given prime

More problems related to Arithmetic Progression

Sum of first n terms of a given series 3, 6, 11, …..
Ratio of mth and nth terms of an A. P. with given ratio of sums
Probability for three randomly chosen numbers to be in AP
Print all triplets in sorted array that form AP
Program for N-th term of Arithmetic Progression series
Sum of Arithmetic Geometric Sequence
Count of AP (Arithmetic Progression) Subsequences in an array

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