ConvertOnlineSum of cubes of first n natural numbers
The sum of the cubes of first n natural numbers is
13 + 23 + 33 + ... + n3
Sum of cubes of first n natural numbers
Calculating the Sum of Cubes of the First n Natural Numbers
The sum of the cubes of the first n natural numbers can be calculated using a specific formula. This formula provides a convenient method to determine the sum without manually calculating the cube of each number and adding them together.
The formula used is:
\(\text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2\)
Where:
- n represents the number of natural numbers.
This can be expressed as:
\( 1^3 + 2^3 + 3^3 + ... + n^3 \)
Examples
Let’s explore some examples to understand how this formula is applied in practice.
1. Calculate the sum of cubes of the first 8 natural numbers.
Answer
To begin, we determine the value of n:
Given:
- n = 8
Next, we apply the standard formula:
Formula:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute the value of n into the formula to calculate the sum:
Substitution:
\( \text{Sum} = \left(\dfrac{8(8+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{8(9)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{72}{2}\right)^2 \)
\( \text{Sum} = (36)^2 \)
\( \text{Sum} = 1296 \)
Finally, we conclude with the result:
Result:
∴ The sum of cubes of the first 8 natural numbers is 1296.
2. What is the sum of the cubes of the first 5 natural numbers?
Answer
We start by determining the value of n:
Given:
- n = 5
Next, we use the established formula to find the sum:
Formula:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute n into the formula and simplify:
Substitution:
\( \text{Sum} = \left(\dfrac{5(5+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{5(6)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{30}{2}\right)^2 \)
\( \text{Sum} = (15)^2 \)
\( \text{Sum} = 225 \)
The result is:
Result:
∴ The sum of cubes of the first 5 natural numbers is 225.
3. Calculate the sum of the cubes 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:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We then substitute n into the formula and perform the calculation:
Substitution:
\( \text{Sum} = \left(\dfrac{10(10+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{10(11)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{110}{2}\right)^2 \)
\( \text{Sum} = (55)^2 \)
\( \text{Sum} = 3025 \)
The final result is:
Result:
∴ The sum of cubes of the first 10 natural numbers is 3025.
4. Find the sum of cubes of the first 12 natural numbers.
Answer
We begin by determining n:
Given:
- n = 12
Next, we use the standard formula to perform the calculation:
Formula:
\( \text{Sum} = \left(\dfrac{ n(n+1)}{2}\right)^2 \)
We substitute the value of n into the formula and simplify:
Substitution:
\( \text{Sum} = \left(\dfrac{12(12+1)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{12(13)}{2}\right)^2 \)
\( \text{Sum} = \left(\dfrac{156}{2}\right)^2 \)
\( \text{Sum} = (78)^2 \)
\( \text{Sum} = 6084 \)
The final sum is:
Result:
∴ The sum of cubes of the first 12 natural numbers is 6084.
Formula
To calculate the sum of cubes of first n natural numbers, you can use the following formula.