Express a number as sum of consecutive numbers

04 Oct, 2021
Given a number N, write a function to express N as sum of two or more consecutive positive numbers. If there is no solution, output -1. If there are multiple solution, then print one of them.Examples:  
Input : N = 10
Output : 4 + 3 + 2 + 1

Input  : N = 8
Output : -1

Input  : N = 24
Output : 9 + 8 + 7
 
 
Sum of first n natural numbers = n * (n + 1)/2

Sum of first (n + k) numbers = (n + k) * (n + k + 1)/2

If N is sum of k consecutive numbers, then
following must be true.

N = [(n+k)(n+k+1) - n(n+1)] / 2

OR 

2 * N = [(n+k)(n+k+1) - n(n+1)]
Below is the implementation based on above idea. 
C++
Java
Python3
C#
PHP
Javascript
// C++ program to print a consecutive sequence// to express N if possible.#include <bits/stdc++.h>using namespace std; // Print consecutive numbers from// last to firstvoid printConsecutive(int last, int first){    cout << first++;    for (int x = first; x<= last; x++)        cout << " + " << x;} void findConsecutive(int N){    for (int last=1; last<N; last++)    {        for (int first=0; first<last; first++)        {            if (2*N == (last-first)*(last+first+1))            {                cout << N << " = ";                printConsecutive(last, first+1);                return;            }        }    }    cout << "-1";} // Driver codeint main(){    int n = 12;    findConsecutive(n);    return 0;}
// Java program to print a consecutive sequence// to express N if possible.import java.util.*;class GFG{ // Print consecutive numbers from// last to firststatic void printConsecutive(int last, int first){    System.out.print(first++);    for (int x = first; x<= last; x++)        System.out.print(" + " +  x);} static void findConsecutive(int N){    for (int last = 1; last < N; last++)    {        for (int first = 0; first < last; first++)        {            if (2*N == (last-first)*(last+first+1))            {                System.out.print(N+ " = ");                printConsecutive(last, first+1);                return;            }        }    }    System.out.print("-1");} // Driver codepublic static void main(String[] args){    int n = 12;    findConsecutive(n);}} // This code is contributed by umadevi9616
# Python3 program to print a consecutive# sequence to express N if possible. # Print consecutive numbers# from last to firstdef printConsecutive(last, first):    print (first, end = "")    first += 1    for x in range(first, last + 1):        print (" +", x, end = "") def findConsecutive(N):    for last in range(1, N):                 for first in range(0, last):                         if 2 * N == (last - first) * (last + first + 1):                print (N, "= ", end = "")                printConsecutive(last, first + 1)                return    print ("-1") # Driver coden = 12findConsecutive(n) # This code is contributed by Shreyanshi Arun.
// C# program to print a consecutive sequence// to express N if possible.using System; class GfG{    // Print consecutive numbers from    // last to first    static void printConsecutive(int last, int first)    {        Console.Write(first++);        for (int x = first; x <= last; x++)            Console.Write(" + "+x);    }         static void findConsecutive(int N)    {        for (int last = 1; last < N; last++)        {            for (int first = 0; first < last; first++)            {                if (2 * N == (last - first)                    * (last + first + 1))                {                    Console.Write(N + " = ");                    printConsecutive(last, first + 1);                    return;                }            }        }        Console.Write("-1");    }         // Driver code    public static void Main ()    {        int n = 12;        findConsecutive(n);    }} // This code is contributed by vt_m
<?php// PHP program to print a consecutive// sequence to express N if possible. // Print consecutive numbers from// last to firstfunction printConsecutive($last, $first){    echo $first++;    for ($x = $first; $x<= $last; $x++)        echo " + " , $x;} function findConsecutive($N){    for ($last = 1; $last < $N; $last++)    {        for ($first = 0; $first < $last; $first++)        {            if (2 * $N == ($last - $first) *                      ($last + $first + 1))            {                 echo $N , " = ";                printConsecutive($last, $first + 1);                return;            }        }    }    echo "-1";}     // Driver Code    $n = 12;    findConsecutive($n);     // This code is contributed by nitin mittal?>
<script>    // Javascript program to print a consecutive// sequence to express N if possible.   // Print consecutive numbers from// last to firstfunction printConsecutive(last, first){    document.write(first++);    for (let x = first; x<= last; x++)        document.write( " + " + x);}   function findConsecutive(N){    for (let last = 1; last < N; last++)    {        for (let first = 0; first < last; first++)        {            if (2 * N == (last - first) *                      (last + first + 1))            {                 document.write(N + " = ");                printConsecutive(last, first + 1);                return;            }        }    }    document.write("-1");}       // Driver Code    let n = 12;    findConsecutive(n);       // This code is contributed by _saurabh_jaiswal</script>
Output:  
12 = 3 + 4 + 5
Reference : https://math.stackexchange.com/questions/139842/in-how-many-ways-can-a-number-be-expressed-as-a-sum-of-consecutive-numbersThis article is contributed by Roshni Agarwal. If you like GeeksforGeeks and would like to contribute, you can also write an article using write.geeksforgeeks.org or mail your article to review-team@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above. 
nitin mittal
_saurabh_jaiswal
umadevi9616
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