ConvertOnlineSum of squares of first n natural numbers
The sum of the squares of first n natural numbers is
12 + 22 + 32 + ... + n2
Sum of squares of first n natural numbers
Calculating the Sum of Squares of the First n Natural Numbers
The sum of the squares of the first n natural numbers can be computed using a well-known arithmetic formula. This formula allows for the quick and efficient calculation of the sum without the need to manually square and add each number individually.
The formula used is:
\(\text{Sum} = \dfrac{ n(n+1)(2n +1)}{6} \)
Where:
- n is the number of natural numbers considered.
This can be expressed as:
\( 1^2 + 2^2 + 3^2 + ... + n^2 \)
Examples
Let’s look at a few examples to understand how this formula is applied in practice.
1. Calculate the sum of squares of the first 8 natural numbers.
Answer
To solve this, we start by identifying the value of n:
Given:
- n = 8
Next, we apply the standard formula:
Formula:
\( \text{Sum} = \dfrac{ n(n+1)(2n +1)}{6} \)
We substitute the value of n into the formula to begin the calculation:
Substitution:
\( \text{Sum} = \dfrac{8(8+1)(2(8) +1)}{6} \)
\( \text{Sum} = \dfrac{8(9)(17)}{6} \)
\( \text{Sum} = \dfrac{1224}{6} \)
\( \text{Sum} = 204 \)
Finally, we conclude with the result:
Result:
∴ The sum of squares of the first 8 natural numbers is 204.
2. What is the sum of the squares of the first 5 natural numbers?
Answer
We begin by determining the value of n:
Given:
- n = 5
Next, we use the established formula:
Formula:
\( \text{Sum} = \dfrac{ n(n+1)(2n +1)}{6} \)
We then substitute n into the formula and simplify:
Substitution:
\( \text{Sum} = \dfrac{5(5+1)(2(5) +1)}{6} \)
\( \text{Sum} = \dfrac{5(6)(11)}{6} \)
\( \text{Sum} = \dfrac{330}{6} \)
\( \text{Sum} = 55 \)
Finally, the result is:
Result:
∴ The sum of squares of the first 5 natural numbers is 55.
3. Calculate the sum of the squares of the first 10 natural numbers.
Answer
We first identify the value of n:
Given:
- n = 10
Next, we apply the formula to calculate the sum:
Formula:
\( \text{Sum} = \dfrac{ n(n+1)(2n +1)}{6} \)
We then substitute n into the formula:
Substitution:
\( \text{Sum} = \dfrac{10(10+1)(2(10) +1)}{6} \)
\( \text{Sum} = \dfrac{10(11)(21)}{6} \)
\( \text{Sum} = \dfrac{2310}{6} \)
\( \text{Sum} = 385 \)
The final result is:
Result:
∴ The sum of squares of the first 10 natural numbers is 385.
4. Find the sum of squares of the first 12 natural numbers.
Answer
We start by determining n:
Given:
- n = 12
Next, we use the standard formula to perform the calculation:
Formula:
\( \text{Sum} = \dfrac{ n(n+1)(2n +1)}{6} \)
We then substitute and simplify the expression:
Substitution:
\( \text{Sum} = \dfrac{12(12+1)(2(12) +1)}{6} \)
\( \text{Sum} = \dfrac{12(13)(25)}{6} \)
\( \text{Sum} = \dfrac{3900}{6} \)
\( \text{Sum} = 650 \)
The final sum is:
Result:
∴ The sum of squares of the first 12 natural numbers is 650.
Formula
To calculate the sum of squares of first n natural numbers, you can use the following formula.