Sum of Squares of First N Numbers Calculator

Welcome to the sum of squares of first N numbers calculator, where the following 5 problems can be easily solved:

Natural numbers refer to the set of non-negative integers, i.e. 0 and positive integers, which are often represented by N. To calculate the sum of squares of first N natural numbers, first, you must find the values of first N natural numbers, then calculate the square of each number, and finally add the squares to get the answer.

For example, the sum of squares of first 5 natural numbers.

1. First 5 natural numbers are: 0, 1, 2, 3, 4
2. The square values of first 5 natural numbers are: 0, 1, 4, 9, 16
3. The sum of squares of first 5 natural numbers is: 0 + 1 + 4 + 9 + 16 = 30

Calculate the sum of squares of first N natural numbers using the formula

From the definition of natural numbers above, we know that natural numbers are a set of arithmetic sequences, the first item is 0, and the tolerance is 1. According to the formula of the sum of squares of first N items of arithmetic sequence

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

Among them, a₁ is the first item of the sequence, d is the tolerance.

The formula of the sum of squares of first N natural numbers can be derived as follows:

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

= n * 0² + n * (n – 1) * 0 * 1 + n * (n – 1) * (2 * n – 1)* 1²6

= n * (n – 1) * (2 * n – 1)6

For example, the sum of squares of first 5 natural numbers above, use the formula to calculate:

5 * (5 – 1) * (2 * 5 – 1)6 = 5 * 4 * 96 = 30

It is consistent with the calculation result above.

A number that can be divided by 2 is called an even number. Note that, the quotient of 0 divided by 2 is also 0, so 0 is also classified as an even number. To calculate the sum of squares of first N even numbers, first, you must find the values of first N even numbers, then calculate the square of each number, and finally add the squares to get the answer.

For example, the sum of squares of first 5 even numbers.

1. First 5 even numbers are: 0, 2, 4, 6, 8
2. The square values of first 5 even numbers are: 0, 4, 16, 36, 64
3. The sum of squares of first 5 even numbers is: 0 + 4 + 16 + 36 + 64 = 120

Calculate the sum of squares of first N even numbers using the formula

Obviously, even numbers are also a set of arithmetic sequences, the first item is 0, and the tolerance is 2. According to the formula of the sum of squares of first N items of arithmetic sequence

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

Among them, a₁ is the first item of the sequence, d is the tolerance.

The formula of the sum of squares of first N even numbers can be derived as follows:

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

= n * 0² + n * (n – 1) * 0 * 1 + n * (n – 1) * (2 * n – 1)* 2²6

= 2 * n * (n – 1) * (2 * n – 1)3

So, the sum of squares of first 5 even numbers can be calculated by the formula:

2 * 5 * (5 – 1) * (2 * 5 – 1)3 = 2 * 5 * 4 * 93 = 120

It is consistent with the calculation result above.

In integers, numbers that cannot be divided by 2 are called odd numbers. Similarly, to calculate the sum of squares of first N odd numbers, first, you must find the values of first N odd numbers, then calculate the square of each number, and finally add the squares to get the answer.

For example, the sum of squares of first 5 odd numbers.

1. First 5 odd numbers are: 1, 3, 5, 7, 9
2. The square values of first 5 odd numbers are: 1, 9, 25, 49, 81
3. The sum of squares of first 5 odd numbers is: 1 + 9 + 25 + 49 + 81 = 165

Calculate the sum of squares of first N odd numbers using the formula

Like even numbers, odd numbers are also a set of arithmetic sequences, the first item is 1, and the tolerance is 2. According to the formula of the sum of squares of first N items of arithmetic sequence

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

Among them, a₁ is the first item of the sequence, d is the tolerance.

The formula of the sum of squares of first N odd numbers can be derived as follows:

S_n = n * a₁² + n * (n – 1) * a₁ * d + n * (n – 1) * (2 * n – 1)* d²6

= n * 1² + n * (n – 1) * 1 * 1 + n * (n – 1) * (2 * n – 1)* 2²6

= n * (2 * n – 1) + 2 * n * (n – 1) * (2 * n – 1)3

So, the sum of squares of first 5 odd numbers can be calculated by the formula:

5 * (2 * 5 – 1) + 2 * 5 * (5 – 1) * (2 * 5 – 1)3 = 5 * 9 + 2 * 5 * 4 * 93 = 165

It is consistent with the calculation result above.

A prime number is an integer that has no factors other than 1 and itself among the natural numbers greater than 1. To calculate the sum of squares of first N prime numbers, first, you must find the values of first N prime numbers, then calculate the square of each number, and finally add the squares to get the answer.

For example, the sum of squares of first 5 prime numbers.

1. First 5 prime numbers are: 2, 3, 5, 7, 11
2. The square values of first 5 prime numbers are: 4, 9, 25, 49, 121
3. The sum of squares of first 5 prime numbers is: 4 + 9 + 25 + 49 + 121 = 208

2² + 3² + 5² + 7² + 11² = 4 + 9 + 25 + 49 + 121 = 208

A composite number is an integer greater than 1 that is divisible by other integers in addition to 1 and itself. To calculate the sum of squares of first N composite numbers, first, you must find the values of first N prime numbers, then calculate the square of each number, and finally add the squares to get the answer.

For example, the sum of squares of first 5 composite numbers.

1. First 5 composite numbers are: 4, 6, 8, 9, 10
2. The square values of first 5 composite numbers are: 16, 36, 64, 81, 100
3. The sum of squares of first 5 composite numbers is: 16 + 36 + 64 + 81 + 100= 297

4² + 6² + 8² + 9² + 10² = 16 + 36 + 64 + 81 + 100= 297

The procedure to use the sum of squares of first N numbers calculator is as follows: