ConvertOnlineSum of first n natural numbers
The sum of first n natural numbers is
1 + 2 + 3 + ... + n
Sum of first n natural numbers
Calculating the Sum of First n Natural Numbers
The sum of the first n natural numbers can be easily calculated using a well-known mathematical formula. Natural numbers are the set of positive integers starting from 1 (i.e., 1, 2, 3, ...).
The formula for calculating the sum of the first n natural numbers is:
\(\text{Sum} = \dfrac{ n ( n + 1)}{2} \)
where:
- n is the total number of natural numbers you want to sum up.
This formula works because it pairs the first and last numbers in the series, which always add up to the same total, and then multiplies that total by the number of such pairs.
Examples
The following examples demonstrate how to calculate the sum of the first n natural numbers using this formula.
1. What is the sum of the first 8 natural numbers?
Answer
Given:
- n = 8
The formula to calculate the sum is:
\(\text{Sum} = \dfrac{ n ( n + 1)}{2} \)
Substituting the given value into the formula:
\(\text{Sum} = \dfrac{ 8 ( 8 + 1)}{2} \)
Simplifying further:
\(\text{Sum} = \dfrac{ 8 \times 9}{2} \)
\(\text{Sum} = \dfrac{72}{2} = 36\)
Therefore, the sum of the first 8 natural numbers is 36.
2. A student wants to calculate the total number of candies if they receive 1 candy on the first day, 2 candies on the second day, and so on, for 10 days. What is the total number of candies?
Answer
Given:
- n = 10 days
The formula to calculate the total number of candies is:
\(\text{Sum} = \dfrac{ n ( n + 1)}{2} \)
Substituting the given value into the formula:
\(\text{Sum} = \dfrac{ 10 ( 10 + 1)}{2} \)
Simplifying further:
\(\text{Sum} = \dfrac{ 10 \times 11}{2} \)
\(\text{Sum} = \dfrac{110}{2} = 55\)
Therefore, the student will receive a total of 55 candies over the 10 days.
3. A staircase has 15 steps, with the number of steps increasing by 1 with each level (i.e., the first level has 1 step, the second level has 2 steps, and so on). What is the total number of steps?
Answer
Given:
- n = 15 levels
The formula to calculate the total number of steps is:
\(\text{Sum} = \dfrac{ n ( n + 1)}{2} \)
Substituting the given value into the formula:
\(\text{Sum} = \dfrac{ 15 ( 15 + 1)}{2} \)
Simplifying further:
\(\text{Sum} = \dfrac{ 15 \times 16}{2} \)
\(\text{Sum} = \dfrac{240}{2} = 120\)
Therefore, the staircase has a total of 120 steps.
4. A builder is arranging bricks in a triangular pattern, where the first row has 1 brick, the second row has 2 bricks, and so on, until the 12th row. How many bricks are needed in total?
Answer
Given:
- n = 12 rows
The formula to calculate the total number of bricks is:
\(\text{Sum} = \dfrac{ n ( n + 1)}{2} \)
Substituting the given value into the formula:
\(\text{Sum} = \dfrac{ 12 ( 12 + 1)}{2} \)
Simplifying further:
\(\text{Sum} = \dfrac{ 12 \times 13}{2} \)
\(\text{Sum} = \dfrac{156}{2} = 78\)
Therefore, the builder needs 78 bricks to complete the triangular pattern.
Formula
To calculate the sum of first n natural numbers, you can use the following formula.