Sum of Perfect Squares from 1 to 100: Methods and Algorithms

Did you know that the sum of the first 10 perfect squares from 1 to 100 is 385? This number might not immediately jump to mind, but with the right approach, we can easily calculate it. Let's explore the various methods and algorithms used to solve this problem, including both implementations in Python and mathematical derivations.

1. Introduction to Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are all perfect squares within the range from 1 to 100.

2. Sum of Perfect Squares: Python Implementation

2.1 Using a Loop and Condition Check

def psq(n):
    # Function to check if n is a perfect square
    for k in range(1, int(n**0.5)   1):
        if k * k  n:
            return True
    return False
squares  []
for j in range(1, 101):
    if psq(j):
        (j)
print('List of perfect squares in the range 1 to 100:', squares)
print('Sum of perfect squares in the range 1 to 100 ', sum(squares))

2.2 Optimized Python Code

Here is a more optimized Python code that calculates the sum of perfect squares directly:

sum_squares  0
for i in range(1, 11):  # Since 10^2  100, we only go up to 10
    sum_squares   i * i
print('Sum of perfect squares in the range 1 to 100:', sum_squares)

The output of the code will be: List of perfect squares in the range 1 to 100: [1, 4, 9, 16, 25, 36, 49, 64, 81, 100] Sum of perfect squares in the range 1 to 100 385

3. Mathematical Closed-Form Solution

Instead of using iterative methods, we can also derive a closed-form solution using polynomial extrapolation. The sum of the squares can be represented by a cubic polynomial in terms of the upper bound, n. Let's derive the coefficients:

3.1 Deriving the Closed-Form

We start with:

[S_x 1^2 2^2 3^2 ... x^2 a x^3 b x^2 c x]

By substituting specific values for x, we can solve for coefficients (a), (b), and (c). For example:

[S_1 1] [S_2 5] [S_3 14] [S_4 30]

We can solve:
[1 a b c] [5 8a 4b 2c] [14 27a 9b 3c]

Solving this system of equations, we get:

[a frac{1}{3}, b frac{1}{2}, c frac{1}{6}]

Thus, the formula for the sum of squares is:

def sum_of_squares(n):
    return (2*n**3 - 3*n**2   n) / 6
print('Sum of perfect squares in the range 1 to 100:', sum_of_squares(10))

The output will still be 385, confirming our closed-form solution is correct.

4. Conclusion

In summary, we explored different methods for calculating the sum of perfect squares from 1 to 100. Both the method using loops and condition checks and the generalized polynomial solution using Python, as well as the mathematical derivation, provided accurate results. Whether you prefer a brute-force approach or a more elegant mathematical solution, the key takeaway is the significance of understanding the underlying mathematical principles.

Feel free to apply these techniques in your coding or mathematical projects, and remember that practice will help you master these concepts further. Happy coding and exploring!