What is the sum of the square of integers from 1 to 100?

The sum of the squares of integers from 1 to 100 is 338350.

This is a common mathematical problem that can be solved using a specific formula. The formula for the sum of the squares of the first 'n' natural numbers is given by:

S = n(n + 1)(2n + 1) / 6

Here's how it applies to our case, where n = 100:

S = 100(100 + 1)(2 * 100 + 1) / 6

S = 100 * 101 * 201 / 6

S = 2030100 / 6

S = 338350

Breakdown of the Calculation

The reference confirms that the sum of the squares of the first 100 natural numbers is 338350. This mathematical concept has applications in various fields like statistics, physics and computer science. Let's break down the key components:

• Integers: These are whole numbers, including positives, negatives, and zero. In our case, we consider the positive integers from 1 to 100.
• Square of an Integer: The square of an integer is the result of multiplying that integer by itself (e.g., 2 squared is 2 * 2 = 4).
• Sum of Squares: This involves squaring each integer in the sequence (1 to 100) and then adding all the results together.

Practical Insight

While it's possible to compute this manually, the formula provides a much quicker and more reliable way to obtain the answer. Understanding this formula not only provides the solution to this specific problem but also helps in solving similar problems involving the sum of squares of a sequence of integers.

Example

If we wanted to calculate the sum of squares for the first 5 integers, we can use the formula as well.

S = 5(5 + 1)(2 * 5 + 1) / 6

S = 5 * 6 * 11 / 6

S = 330 / 6

S = 55

Manually:

1² + 2² + 3² + 4² + 5² = 1 + 4 + 9 + 16 + 25 = 55

As can be seen, both methods provided the same correct answer.