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++

`// 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 first` `void` `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 code` `int` `main()` `{` ` ` `int` `n = 12;` ` ` `findConsecutive(n);` ` ` `return` `0;` `}` |

Java

```
// Java program to print a consecutive sequence
// to express N if possible.
class GfG
{
// Print consecutive numbers from
// last to first
static 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.printf(N + " = ");
printConsecutive(last, first+1);
return;
}
}
}
System.out.print("-1");
}
// main function
public static void main (String[] args)
{
int n = 12;
findConsecutive(n);
}
}
```

## Python3

`# Python3 program to print a consecutive` `# sequence to express N if possible.` `# Print consecutive numbers` `# from last to first` `def` `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 code` `n ` `=` `12` `findConsecutive(n)` `# This code is contributed by Shreyanshi Arun.` |

## C#

`// 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` `// PHP program to print a consecutive` `// sequence to express N if possible.` `// Print consecutive numbers from` `// last to first` `function` `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` `?>` |

## Javascript

`<script>` ` ` `// Javascript program to print a consecutive` `// sequence to express N if possible.` ` ` `// Print consecutive numbers from` `// last to first` `function` `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-numbers

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