前n个自然数的平方和之和-yiteyi-C++库

给定一个正整数 N .任务是求前n个自然数的平方和之和。 例如：

null

Input : n = 3Output : 20Sum of square of first natural number = 1Sum of square of first two natural number = 1^2 + 2^2 = 5Sum of square of first three natural number = 1^2 + 2^2 + 3^2 = 14Sum of sum of square of first three natural number = 1 + 5 + 14 = 20Input : n = 2Output : 6

推荐：请尝试你的方法 {IDE} 首先，在进入解决方案之前。

方法1:O（n） 其思想是求第一个i自然数的平方和，其中1<=i<=n，然后将它们相加。我们可以通过以下公式求出前n个自然数的平方和：n*（n+1）*（2*n+1）/6 以下是该方法的实施情况：

C++

                         // CPP Program to find the sum of sum of                       
                         // squares of first n natural number                       
                         #include <bits/stdc++.h>                       
                         using                                     namespace                                     std;                       
           
                         // Function to find sum of sum of square of                       
                         // first n natural number                       
                         int                                     findSum(                                     int                                     n)                       
                         {                       
                                                 int                                     sum = 0;                       
                                                 for                                     (                                     int                                     i = 1; i <= n; i++)                       
                                                 sum += ((i * (i + 1) * (2 * i + 1)) / 6);                       
                                                 return                                     sum;                       
                         }                       
           
                         // Driven Program                       
                         int                                     main()                       
                         {                       
                                                 int                                     n = 3;                       
                                                 cout << findSum(n) << endl;                       
                                                 return                                     0;                       
                         }                       

JAVA

                         // Java Program to find the sum of                       
                         // sum of squares of first n natural                       
                         // number                       
           
                         class                                     GFG {                       
                                   
                                                 // Function to find sum of sum of                       
                                                 // square of first n natural number                       
                                                 static                                     int                                     findSum(                                     int                                     n)                       
                                                 {                       
                                                 int                                     sum =                                     0                                     ;                       
                                                 for                                     (                                     int                                     i =                                     1                                     ; i <= n; i++)                       
                                                 sum += ((i * (i +                                     1                                     )                       
                                                 * (                                     2                                     * i +                                     1                                     )) /                                     6                                     );                       
                                                 return                                     sum;                       
                                                 }                       
                                   
                                                 // Driver Program                       
                                                 public                                     static                                     void                                     main(String[] args)                       
                                                 {                       
                                                 int                                     n =                                     3                                     ;                       
                                   
                                                 System.out.println( findSum(n));                       
                                                 }                       
                         }                       
           
           
                         // This code is contributed by                       
                         // Arnab Kundu                       

Python3

                         # Python3 Program to find the sum                       
                         # of sum of squares of first n                       
                         # natural number                       
           
                         # Function to find sum of sum of                       
                         # square of first n natural number                       
                         def                                     findSum(n):                       
                                                 summ                                     =                                     0                       
                                                 for                                     i                                     in                                     range                                     (                                     1                                     , n                                     +                                     1                                     ):                       
                                                 summ                                     =                                     (summ                                     +                                     ((i                                     *                                     (i                                     +                                     1                                     )                       
                                                 *                                     (                                     2                                     *                                     i                                     +                                     1                                     ))                                     /                                     6                                     ))                       
                                                 return                                     summ                       
           
                         # Driven Program                       
                         n                                     =                                     3                       
                         print                                     (                                     int                                     (findSum(n)))                       
           
           
                         # This code is contributed by                       
                         # Prasad Kshirsagar                       

C#

                         // C# Program to find the sum of sum of                       
                         // squares of first n natural number                       
                         using                                     System;                       
           
                         public                                     class                                     GFG {                       
                                   
                                                 // Function to find sum of sum of                       
                                                 // square of first n natural number                       
                                                 static                                     int                                     findSum(                                     int                                     n)                       
                                                 {                       
                                                 int                                     sum = 0;                       
                                                 for                                     (                                     int                                     i = 1; i <= n; i++)                       
                                                 sum += ((i * (i + 1) *                       
                                                 (2 * i + 1)) / 6);                       
                                                 return                                     sum;                       
                                                 }                       
                                   
                                                 // Driver Program                       
                                                 static                                     public                                     void                                     Main()                       
                                                 {                       
                                                 int                                     n = 3;                       
                                   
                                                 Console.WriteLine(findSum(n));                       
                                                 }                       
                         }                       
           
                         // This code is contributed by                       
                         // Arnab Kundu.                       

PHP

                         <?php                       
                         // PHP Program to find the sum of                       
                         // squares of first n natural number                       
           
                         // Function to find sum of sum of                       
                         // square of first n natural number                       
                         function                                     findSum(                                     $n                                     )                       
                         {                       
                                                 $sum                                     = 0;                       
                                                 for                                     (                                     $i                                     = 1;                                     $i                                     <=                                     $n                                     ;                                     $i                                     ++)                       
                                                 $sum                                     += ((                                     $i                                     * (                                     $i                                     + 1) *                       
                                                 (2 *                                     $i                                     + 1)) / 6);                       
                                                 return                                     $sum                                     ;                       
                         }                       
           
                         // Driver Code                       
                         $n                                     = 3;                       
                         echo                                     findSum(                                     $n                                     ) ;                       
           
                         // This code is contributed by anuj_67.                       
                         ?>                       

Javascript

                         <script>                       
           
                         // Javascript Program to find the sum of sum of                       
                         // squares of first n natural number                       
           
                         // Function to find sum of sum of square                       
                         // of first n natural number                       
                         function                                     findSum(n)                       
                         {                       
                                                 return                                     (n * (n + 1) * (n + 1) * (n + 2)) / 12;                       
                         }                       
           
                         // Driven Program                       
           
                                                 let n = 3;                       
                                                 document.write(findSum(n) +                                     "<br>"                                     );                       
           
           
                         // This code is contributed by Mayank Tyagi                       
           
                         </script>                       

输出：

时间复杂度：O（n）辅助空间：O（1） 方法2:O（1） 数学上，我们需要找到，∑（（i*（i+1）*（2*i+1）/6），其中1<=i<=n 让我们来解决这个求和问题，

Sum = Σ ((i * (i + 1) * (2*i + 1)/6), where 1 <= i <= n    = (1/6)*(Σ ((i * (i + 1) * (2*i + 1)))    = (1/6)*(Σ ((i² + i) * (2*i + 1))    = (1/6)*(Σ ((2*i³ + 3*i² + i))    = (1/6)*(Σ 2*i³ + Σ 3*i² + Σ i)    = (1/6)*(2*Σ i³ + 3*Σ i² + Σ i)    = (1/6)*(2*(i*(i + 1)/2)² + 3*(i * (i + 1) * (2*i + 1))/6 + i * (i + 1)/2)    = (1/6)*(i * (i + 1))((i*(i + 1)/2) + ((2*i + 1)/2) + 1/2)    = (1/12) * (i * (i + 1)) * ((i*(i + 1)) + (2*i + 1) + 1)    = (1/12) * (i * (i + 1)) * ((i² + 3 * i + 2)    = (1/12) * (i * (i + 1)) * ((i + 1) * (i + 2))    = (1/12) * (i * (i + 1)² * (i + 2))

以下是该方法的实施情况：

C++

                         // CPP Program to find the sum of sum of                       
                         // squares of first n natural number                       
                         #include <bits/stdc++.h>                       
                         using                                     namespace                                     std;                       
           
                         // Function to find sum of sum of square                       
                         // of first n natural number                       
                         int                                     findSum(                                     int                                     n)                       
                         {                       
                                                 return                                     (n * (n + 1) * (n + 1) * (n + 2)) / 12;                       
                         }                       
           
                         // Driven Program                       
                         int                                     main()                       
                         {                       
                                                 int                                     n = 3;                       
                                                 cout << findSum(n) << endl;                       
                                                 return                                     0;                       
                         }                       

JAVA

                         // Java Program to find the sum of sum of                       
                         // squares of first n natural number                       
                         class                                     GFG {                       
           
                                                 // Function to find sum of sum of                       
                                                 // square of first n natural number                       
                                                 static                                     int                                     findSum(                                     int                                     n)                       
                                                 {                       
                                                 return                                     (n * (n +                                     1                                     ) *                       
                                                 (n +                                     1                                     ) * (n +                                     2                                     )) /                                     12                                     ;                       
                                                 }                       
                                   
                                                 // Driver Program                       
                                                 public                                     static                                     void                                     main(String[] args)                       
                                                 {                       
                                                 int                                     n =                                     3                                     ;                       
                                   
                                                 System.out.println(findSum(n) );                       
                                                 }                       
                         }                       
           
                         // This code is contributed by Arnab Kundu                       

Python3

                         # Python3 Program to find the sum                       
                         # of sum of squares of first n                       
                         # natural number                       
           
                         # Function to find sum of sum of                       
                         # square of first n natural number                       
                         def                                     findSum(n):                       
                                                 return                                     ((n                                     *                                     (n                                     +                                     1                                     )                                     *                                     (n                                     +                                     1                                     )                       
                                                 *                                     (n                                     +                                     2                                     ))                                     /                                     12                                     )                       
           
                         # Driven Program                       
                         n                                     =                                     3                       
                         print                                     (                                     int                                     (findSum(n)))                       
           
                         # This code is contributed by                       
                         # Prasad Kshirsagar                       

C#

                         // C# Program to find the sum of sum of                       
                         // squares of first n natural number                       
                         using                                     System;                       
           
                         class                                     GFG {                       
           
                                                 // Function to find sum of sum of                       
                                                 // square of first n natural number                       
                                                 static                                     int                                     findSum(                                     int                                     n)                       
                                                 {                       
                                                 return                                     (n * (n + 1) * (n + 1)                       
                                                 * (n + 2)) / 12;                       
                                                 }                       
                                   
                                                 // Driver Program                       
                                                 static                                     public                                     void                                     Main()                       
                                                 {                       
                                                 int                                     n = 3;                       
                                   
                                                 Console.WriteLine(findSum(n) );                       
                                                 }                       
                         }                       
           
                         // This code is contributed by Arnab Kundu                       

PHP

                         <?php                       
                         // PHP Program to find the sum of sum of                       
                         // squares of first n natural number                       
           
                         // Function to find sum of sum of square                       
                         // of first n natural number                       
                         function                                     findSum(                                     $n                                     )                       
                         {                       
                                                 return                                     (                                     $n                                     * (                                     $n                                     + 1) *                       
                                                 (                                     $n                                     + 1) * (                                     $n                                     +                       
                                                 2)) / 12;                       
                         }                       
           
                                                 // Driver Code                       
                                                 $n                                     = 3;                       
                                                 echo                                     findSum(                                     $n                                     ) ;                       
           
                         // This code is contributed by nitin mittal                       
                         ?>                       

Javascript

                         <script>                       
                         // js Program to find the sum of sum of                       
                         // squares of first n natural number                       
           
                         // Function to find sum of sum of square                       
                         // of first n natural number                       
                         function                                     findSum($n)                       
                         {                       
                                                 return                                     (n * (n + 1) *(n + 1) * (n + 2)) / 12;                       
                         }                       
           
                                                 // Driver Code                       
                                                 n = 3;                       
                                                 document.write(findSum(n)) ;                       
           
                         // This code is contributed by sravan kumar                       
                         </script>                       

输出：

时间复杂度：O（1）辅助空间：O（1）

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