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Find the missing number in Arithmetic Progression
  • Difficulty Level : Medium
  • Last Updated : 18 Jun, 2021

Given an array that represents elements of arithmetic progression in order. One element is missing in the progression, find the missing number. 
Examples: 
 

Input: arr[]  = {2, 4, 8, 10, 12, 14}
Output: 6

Input: arr[]  = {1, 6, 11, 16, 21, 31};
Output: 26

 

A Simple Solution is to linearly traverse the array and find the missing number. Time complexity of this solution is O(n). 
We can solve this problem in O(Logn) time using Binary Search. The idea is to go to the middle element. Check if the difference between middle and next to middle is equal to diff or not, if not then the missing element lies between mid and mid+1. If the middle element is equal to n/2th term in Arithmetic Series (Let n be the number of elements in input array), then missing element lies in right half. Else element lies in left half.
Following is implementation of above idea. 
 

C++




// C++ program to find the missing number
// in a given arithmetic progression
#include<iostream>
using namespace std;
#define INT_MAX 2147483647;
class GFG
{
     
// A binary search based recursive function that returns
// the missing element in arithmetic progression
public:int findMissingUtil(int arr[], int low,
                           int high, int diff)
{
    // There must be two elements to find the missing
    if (high <= low)
        return INT_MAX;
 
    // Find index of middle element
    int mid = low + (high - low) / 2;
 
    // The element just after the middle element is missing.
    // The arr[mid+1] must exist, because we return when
    // (low == high) and take floor of (high-low)/2
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // The element just before mid is missing
    if (mid > 0 && arr[mid] - arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // If the elements till mid follow AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissingUtil(arr, mid + 1,
                               high, diff);
 
    // Else recur for left half
    return findMissingUtil(arr, low, mid - 1, diff);
}
 
// The function uses findMissingUtil() to
// find the missing element in AP.
// It assumes that there is exactly one
// missing element and may give incorrect result
// when there is no missing element or
// more than one missing elements. This function
// also assumes that the difference in AP is an
// integer.
int findMissing(int arr[], int n)
{
    // If exactly one element is missing, then we can find
    // difference of arithmetic progression using following
    // formula. Example, 2, 4, 6, 10, diff = (10-2)/4 = 2.
    // The assumption in formula is that the difference is
    // an integer.
    int diff = (arr[n - 1] - arr[0]) / n;
 
    // Binary search for the missing
    // number using above calculated diff
    return findMissingUtil(arr, 0, n - 1, diff);
}
};
 
// Driver Code
int main()
{
    GFG g;
    int arr[] = {2, 4, 8, 10, 12, 14};
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << "The missing element is "
         << g.findMissing(arr, n);
    return 0;
}
 
// This code is contributed by Soumik

C




// A C program to find the missing number in a given
// arithmetic progression
#include <stdio.h>
#include <limits.h>
 
// A binary search based recursive function that returns
// the missing element in arithmetic progression
int findMissingUtil(int arr[], int low, int high, int diff)
{
    // There must be two elements to find the missing
    if (high <= low)
        return INT_MAX;
 
    // Find index of middle element
    int mid = low + (high - low)/2;
 
    // The element just after the middle element is missing.
    // The arr[mid+1] must exist, because we return when
    // (low == high) and take floor of (high-low)/2
    if (arr[mid+1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // The element just before mid is missing
    if (mid > 0 && arr[mid] - arr[mid-1] != diff)
        return (arr[mid-1] + diff);
 
    // If the elements till mid follow AP, then recur
    // for right half
    if (arr[mid] == arr[0] + mid*diff)
        return findMissingUtil(arr, mid+1, high, diff);
 
    // Else recur for left half
    return findMissingUtil(arr, low, mid-1, diff);
}
 
// The function uses findMissingUtil() to find the missing
// element in AP.  It assumes that there is exactly one missing
// element and may give incorrect result when there is no missing
// element or more than one missing elements.
// This function also assumes that the difference in AP is an
// integer.
int findMissing(int arr[], int n)
{
    // If exactly one element is missing, then we can find
    // difference of arithmetic progression using following
    // formula.  Example, 2, 4, 6, 10, diff = (10-2)/4 = 2.
    // The assumption in formula is that the difference is
    // an integer.
    int diff = (arr[n-1] - arr[0])/n;
 
    // Binary search for the missing number using above
    // calculated diff
    return findMissingUtil(arr, 0, n-1, diff);
}
 
/* Driver program to check above functions */
int main()
{
    int arr[] = {2, 4, 8, 10, 12, 14};
    int n = sizeof(arr)/sizeof(arr[0]);
    printf("The missing element is %d", findMissing(arr, n));
    return 0;
}

Java




// A Java program to find
// the missing number in
// a given arithmetic
// progression
import java.io.*;
 
class GFG
{
     
// A binary search based
// recursive function that
// returns the missing
// element in arithmetic
// progression
static int findMissingUtil(int arr[], int low,
                           int high, int diff)
{
    // There must be two elements
    // to find the missing
    if (high <= low)
        return Integer.MAX_VALUE;
 
    // Find index of
    // middle element
    int mid = low + (high - low) / 2;
 
    // The element just after the
    // middle element is missing.
    // The arr[mid+1] must exist,
    // because we return when
    // (low == high) and take
    // floor of (high-low)/2
    if (arr[mid + 1] - arr[mid] != diff)
        return (arr[mid] + diff);
 
    // The element just
    // before mid is missing
    if (mid > 0 && arr[mid] -
                   arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // If the elements till mid follow
    // AP, then recur for right half
    if (arr[mid] == arr[0] + mid * diff)
        return findMissingUtil(arr, mid + 1,
                               high, diff);
 
    // Else recur for left half
    return findMissingUtil(arr, low, mid - 1, diff);
}
 
// The function uses findMissingUtil()
// to find the missing element in AP.
// It assumes that there is exactly
// one missing element and may give
// incorrect result when there is no
// missing element or more than one
// missing elements. This function also
// assumes that the difference in AP is
// an integer.
static int findMissing(int arr[], int n)
{
    // If exactly one element is missing,
    // then we can find difference of
    // arithmetic progression using
    // following formula. Example, 2, 4,
    // 6, 10, diff = (10-2)/4 = 2.
    // The assumption in formula is that
    // the difference is an integer.
    int diff = (arr[n - 1] - arr[0]) / n;
 
    // Binary search for the missing
    // number using above calculated diff
    return findMissingUtil(arr, 0, n - 1, diff);
}
 
// Driver Code
public static void main (String[] args)
{
    int arr[] = {2, 4, 8, 10, 12, 14};
    int n = arr.length;
    System.out.println("The missing element is "+  
                            findMissing(arr, n));
}
}
 
// This code is contributed by anuj_67.

Python3




# A Python3 program to find the missing
# number in a given arithmetic progression
import sys
 
# A binary search based recursive function
# that returns the missing element in
# arithmetic progression
def findMissingUtil(arr, low, high, diff):
 
    # There must be two elements to
    # find the missing
    if (high <= low):
        return sys.maxsize;
 
    # Find index of middle element
    mid = int(low + (high - low) / 2);
 
    # The element just after the middle
    # element is missing. The arr[mid+1]
    # must exist, because we return when
    # (low == high) and take floor of
    # (high-low)/2
    if (arr[mid + 1] - arr[mid] != diff):
        return (arr[mid] + diff);
 
    # The element just before mid is missing
    if (mid > 0 and arr[mid] -
                    arr[mid - 1] != diff):
        return (arr[mid - 1] + diff);
 
    # If the elements till mid follow AP,
    # then recur for right half
    if (arr[mid] == arr[0] + mid * diff):
        return findMissingUtil(arr, mid + 1,
                                high, diff);
 
    # Else recur for left half
    return findMissingUtil(arr, low,
                           mid - 1, diff);
 
# The function uses findMissingUtil() to find
# the missing element in AP. It assumes that
# there is exactly one missing element and may
# give incorrect result when there is no missing
# element or more than one missing elements.
# This function also assumes that the difference
# in AP is an integer.
def findMissing(arr, n):
 
    # If exactly one element is missing, then
    # we can find difference of arithmetic
    # progression using following formula.
    # Example, 2, 4, 6, 10, diff = (10-2)/4 = 2.
    # The assumption in formula is that the
    # difference is an integer.
    diff = int((arr[n - 1] - arr[0]) / n);
 
    # Binary search for the missing number
    # using above calculated diff
    return findMissingUtil(arr, 0, n - 1, diff);
 
# Driver Code
arr = [2, 4, 8, 10, 12, 14];
n = len(arr);
print("The missing element is",
          findMissing(arr, n));
 
# This code is contributed by chandan_jnu

C#




// A C# program to find
// the missing number in
// a given arithmetic
// progression
using System;
 
class GFG
{
     
// A binary search based
// recursive function that
// returns the missing
// element in arithmetic
// progression
static int findMissingUtil(int []arr, int low,
                           int high, int diff)
{
    // There must be two elements
    // to find the missing
    if (high <= low)
        return int.MaxValue;
 
    // Find index of
    // middle element
    int mid = low + (high -
                     low) / 2;
 
    // The element just after the
    // middle element is missing.
    // The arr[mid+1] must exist,
    // because we return when
    // (low == high) and take
    // floor of (high-low)/2
    if (arr[mid + 1] -
        arr[mid] != diff)
        return (arr[mid] + diff);
 
    // The element just
    // before mid is missing
    if (mid > 0 && arr[mid] -
                   arr[mid - 1] != diff)
        return (arr[mid - 1] + diff);
 
    // If the elements till mid follow
    // AP, then recur for right half
    if (arr[mid] == arr[0] +
               mid * diff)
        return findMissingUtil(arr, mid + 1,
                               high, diff);
 
    // Else recur for left half
    return findMissingUtil(arr, low,
                           mid - 1, diff);
}
 
// The function uses findMissingUtil()
// to find the missing element
// in AP. It assumes that there
// is exactly one missing element
// and may give incorrect result
// when there is no missing element
// or more than one missing elements.
// This function also assumes that
// the difference in AP is an integer.
static int findMissing(int []arr, int n)
{
    // If exactly one element
    // is missing, then we can
    // find difference of arithmetic
    // progression using following
    // formula. Example, 2, 4, 6, 10,
    // diff = (10-2)/4 = 2.The assumption
    // in formula is that the difference
    // is an integer.
    int diff = (arr[n - 1] -
                arr[0]) / n;
 
    // Binary search for the
    // missing number using
    // above calculated diff
    return findMissingUtil(arr, 0,
                           n - 1, diff);
}
 
// Driver Code
public static void Main ()
{
    int []arr = {2, 4, 8,
                 10, 12, 14};
    int n = arr.Length;
    Console.WriteLine("The missing element is "+
                           findMissing(arr, n));
}
}
 
// This code is contributed by anuj_67.

PHP




<?php
// A PHP program to find the missing
// number in a given arithmetic progression
 
// A binary search based recursive function
// that returns the missing element in
// arithmetic progression
 
function findMissingUtil($arr, $low, $high, $diff)
{
    // There must be two elements to
    // find the missing
    if ($high <= $low)
        return PHP_INT_MAX;
 
    // Find index of middle element
    $mid = $low + ($high - $low) / 2;
 
    // The element just after the middle
    // element is missing. The arr[mid+1]
    // must exist, because we return when
    // (low == high) and take floor of (high-low)/2
    if ($arr[$mid + 1] - $arr[$mid] != $diff)
        return ($arr[$mid] + $diff);
 
    // The element just before mid is missing
    if ($mid > 0 && $arr[$mid] - $arr[$mid - 1] != $diff)
        return ($arr[$mid - 1] + $diff);
 
    // If the elements till mid follow AP,
    // then recur for right half
    if ($arr[$mid] == $arr[0] + $mid * $diff)
        return findMissingUtil($arr, $mid + 1,
                               $high, $diff);
 
    // Else recur for left half
    return findMissingUtil($arr, $low, $mid - 1, $diff);
}
 
// The function uses findMissingUtil() to find
// the missing element in AP. It assumes that
// there is exactly one missing element and may
// give incorrect result when there is no missing
// element or more than one missing elements.
// This function also assumes that the difference
// in AP is an integer.
function findMissing($arr, $n)
{
    // If exactly one element is missing, then
    // we can find difference of arithmetic
    // progression using following formula.
    // Example, 2, 4, 6, 10, diff = (10-2)/4 = 2.
    // The assumption in formula is that the
    // difference is an integer.
    $diff = ($arr[$n - 1] - $arr[0]) / $n;
 
    // Binary search for the missing number
    //  using above calculated diff
    return findMissingUtil($arr, 0, $n - 1, $diff);
}
 
// Driver Code
$arr = array(2, 4, 8, 10, 12, 14);
$n = sizeof($arr);
echo "The missing element is ",
         findMissing($arr, $n);
 
// This code is contributed by Sach_Code
?>

Javascript




<script>
 
    // A JavaScript program to find
    // the missing number in
    // a given arithmetic
    // progression
     
    // A binary search based
    // recursive function that
    // returns the missing
    // element in arithmetic
    // progression
    function findMissingUtil(arr, low, high, diff)
    {
        // There must be two elements
        // to find the missing
        if (high <= low)
            return Number.MAX_VALUE;
 
        // Find index of
        // middle element
        let mid = low + parseInt((high - low) / 2, 10);
 
        // The element just after the
        // middle element is missing.
        // The arr[mid+1] must exist,
        // because we return when
        // (low == high) and take
        // floor of (high-low)/2
        if (arr[mid + 1] - arr[mid] != diff)
            return (arr[mid] + diff);
 
        // The element just
        // before mid is missing
        if (mid > 0 && arr[mid] -
                       arr[mid - 1] != diff)
            return (arr[mid - 1] + diff);
 
        // If the elements till mid follow
        // AP, then recur for right half
        if (arr[mid] == arr[0] + mid * diff)
            return findMissingUtil(arr, mid + 1, high, diff);
 
        // Else recur for left half
        return findMissingUtil(arr, low, mid - 1, diff);
    }
 
    // The function uses findMissingUtil()
    // to find the missing element in AP.
    // It assumes that there is exactly
    // one missing element and may give
    // incorrect result when there is no
    // missing element or more than one
    // missing elements. This function also
    // assumes that the difference in AP is
    // an integer.
    function findMissing(arr, n)
    {
        // If exactly one element is missing,
        // then we can find difference of
        // arithmetic progression using
        // following formula. Example, 2, 4,
        // 6, 10, diff = (10-2)/4 = 2.
        // The assumption in formula is that
        // the difference is an integer.
        let diff = parseInt((arr[n - 1] - arr[0]) / n, 10);
 
        // Binary search for the missing
        // number using above calculated diff
        return findMissingUtil(arr, 0, n - 1, diff);
    }
     
    let arr = [2, 4, 8, 10, 12, 14];
    let n = arr.length;
    document.write("The missing element is "+  
                            findMissing(arr, n));
   
</script>

Output:  

The missing element is 6

 



Exercise: 
Solve the same problem for Geometrical Series. What is the time complexity of your solution? What about Fibonacci Series?
This article is contributed by Harshit Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
 

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