## Summation of Squares of n Natural Numbers

Summation of Squares of n Natural Numbers

## Summation of Squares of n Natural Numbers

The sum of squares of n natural numbers can be found using the formula:

1^2 + 2^2 + 3^2 + … + n^2 = n(n+1)(2n+1)/6

This formula can be proved using mathematical induction, but for the sake of brevity, we will skip the proof and focus on its application.

Let’s take an example to understand how this formula works.

Example: Find the sum of squares of the first 5 natural numbers.

Solution:

We can use the formula mentioned above to find the sum of squares of the first 5 natural numbers as follows:

1^2 + 2^2 + 3^2 + 4^2 + 5^2 = 5(5+1)(2(5)+1)/6
= 5(6)(11)/6
= 55

Therefore, the sum of squares of the first 5 natural numbers is 55.

This formula can be useful in solving problems in various areas of mathematics such as algebra, calculus, and geometry. For example, it can be used to find the area under a curve, the volume of a solid of revolution, and the length of a curve.

Another interesting application of this formula is in physics, where it is used to calculate the mean square displacement of particles in a random walk. This is because the mean square displacement is given by the sum of squares of the distances traveled by the particle at each step.

Formula for Sum of Squares of n Natural Numbers

The formula for calculating the sum of squares of the first n natural numbers can be derived using mathematical induction. The base case is when n = 1, which gives:

S1 = 1^2 = 1

For the induction step, we assume that the formula holds for n = k, and we want to show that it holds for n = k+1. We can write:

S(k+1) = 1^2 + 2^2 + 3^2 + … + k^2 + (k+1)^2

= Sk + (k+1)^2

Using the formula for Sk, we get:

S(k+1) = (k(k+1)(2k+1))/6 + (k+1)^2

= (k^2 + 2k + 1)((k/6) + 1)

= ((k+1)(k+2)(2k+3))/6

Therefore, the formula for the sum of squares of the first n natural numbers is:

Sn = 1^2 + 2^2 + 3^2 + … + n^2 = (n(n+1)(2n+1))/6

Properties of Sum of Squares of n Natural Numbers

The sum of squares of n natural numbers has several interesting properties. Some of these are:

1. Symmetry: The sum of squares of the first n natural numbers is symmetric about the middle term. That is, S(n+1) – Sn = (n+1)^2.

2. Recurrence relation: The sum of squares of the first n natural numbers can be expressed as a recurrence relation, which is given by:

Sn = Sn-1 + n^2

with S0 = 0.

1. Odd and even numbers: The sum of squares of the first n odd numbers is given by (2n^3 + 3n^2 + n)/3, while the sum of squares of the first n even numbers is given by (2n^3 + 3n^2 + n)/2.

Applications of Sum of Squares of n Natural Numbers

The sum of squares of n natural numbers finds application in many areas of mathematics and science. Some of these are:

1. Statistics: The sum of squares of deviations from the mean is an important concept in statistics. It is used to measure the variation or dispersion of a set of data.

2. Physics: The sum of squares of the components of a vector is equal to the square of its magnitude. This is known as the Pythagorean theorem and is used in physics to calculate distances, forces, and velocities.

3. Engineering: The sum of squares of errors is an important concept in engineering. It is used to measure the accuracy and precision of a system or process.

Here are some examples of Summation of Squares of n Natural Numbers:

Example 1: Find the value of 1^2 + 2^2 + 3^2 + 4^2. Solution: Using the formula, we have: n(n+1)(2n+1)/6 Substituting n = 4 in the formula, we get: 4(4+1)(2*4+1)/6 = 30 Hence, 1^2 + 2^2 + 3^2 + 4^2 = 30.

Example 2: Calculate the value of 1^2 + 2^2 + 3^2 + … + 10^2. Solution: Using the formula, we have: n(n+1)(2n+1)/6 Substituting n = 10 in the formula, we get: 10(10+1)(2*10+1)/6 = 385 Hence, 1^2 + 2^2 + 3^2 + … + 10^2 = 385.

Example 3: Find the value of 5^2 + 6^2 + 7^2 + … + 20^2. Solution: Using the formula, we have: n(n+1)(2n+1)/6 Substituting n = 20 and m = 4 in the formula, we get: 20(20+1)(220+1)/6 – 4(4+1)(24+1)/6 = 11440 – 30 = 11410 Hence, 5^2 + 6^2 + 7^2 + … + 20^2 = 11410.

In conclusion, the sum of squares of n natural numbers is a common mathematical concept that has numerous applications in various fields. The formula for the sum of squares of n natural numbers can be easily derived and is a powerful tool for solving problems in mathematics and physics.