Nicomachus’s Theorem
Nicomachus’s Theorem states that sum of cubes of first n natural numbers is equal to squares of natural number sum.
In other words
Or we can say that the sum is equal to square of n-th triangular number.
Mathematical Induction based proof can be found here.
C++
// CPP program to verify Nicomachus's Theorem #include <bits/stdc++.h> using namespace std; void NicomachusTheorem_sum( int n) { // Compute sum of cubes int sum = 0; for ( int k=1; k<=n; k++) sum += k*k*k; // Check if sum is equal to // given formula. int triNo = n*(n+1)/2; if (sum == triNo * triNo) cout << "Yes" ; else cout << "No" ; } // driver function int main() { int n = 5; NicomachuTheorem_sum(n); return 0; } |
Java
// Java program to verify Nicomachus's Theorem import java.io.*; class GFG { static void NicomachuTheorem_sum( int n) { // Compute sum of cubes int sum = 0 ; for ( int k = 1 ; k <= n; k++) sum += k * k * k; // Check if sum is equal to // given formula. int triNo = n * (n + 1 ) / 2 ; if (sum == triNo * triNo) System.out.println( "Yes" ); else System.out.println( "No" ); } // driver function public static void main (String[] args) { int n = 5 ; NicomachuTheorem_sum(n); } } // This code is contributed by anuj_67. |
Python3
# Python3 program to verify # Nicomachus's Theorem def NicomachuTheorem_sum(n): # Compute sum of cubes sum = 0 ; for k in range ( 1 , n + 1 ): sum + = k * k * k; # Check if sum is equal to # given formula. triNo = n * (n + 1 ) / 2 ; if ( sum = = triNo * triNo): print ( "Yes" ); else : print ( "No" ); # Driver Code n = 5 ; NicomachuTheorem_sum(n); # This code is contributed # by mits |
C#
// C# program to verify // Nicomachus's Theorem using System; class GFG { static void NicomachuTheorem_sum( int n) { // Compute sum of cubes int sum = 0; for ( int k = 1; k <= n; k++) sum += k * k * k; // Check if sum is equal to // given formula. int triNo = n * (n + 1) / 2; if (sum == triNo * triNo) Console.WriteLine( "Yes" ); else Console.WriteLine( "No" ); } // Driver Code public static void Main () { int n = 5; NicomachuTheorem_sum(n); } } // This code is contributed by anuj_67 |
PHP
<?php // PHP program to verify // Nicomachus's Theorem function NicomachuTheorem_sum( $n ) { // Compute sum of cubes $sum = 0; for ( $k = 1; $k <= $n ; $k ++) $sum += $k * $k * $k ; // Check if sum is equal to // given formula. $triNo = $n * ( $n + 1) / 2; if ( $sum == $triNo * $triNo ) echo "Yes" ; else echo "No" ; } // Driver Code $n = 5; NicomachuTheorem_sum( $n ); // This code is contributed by anuj_67. ?> |
Javascript
<script> // JavaScript program to verify Nicomachus's Theorem function NicomachuTheorem_sum(n) { // Compute sum of cubes let sum = 0; for (let k = 1; k <= n; k++) sum += k * k * k; // Check if sum is equal to // given formula. let triNo = n * (n + 1) / 2; if (sum == triNo * triNo) document.write( "Yes" ); else document.write( "No" ); } // Driver code let n = 5; NicomachuTheorem_sum(n); // This code is contributed by souravghosh0416. </script> |
Output:
Yes
Time complexity : O(n)
Auxiliary Space : O(1)