What is the average of the first 57 natural numbers?
Understand the Problem
The question is asking for the average value of the first 57 natural numbers. To find this, we will sum the first 57 natural numbers and then divide by 57 to get the average.
Answer
The average of the first 57 natural numbers is $29$.
Answer for screen readers
The average value of the first 57 natural numbers is $29$.
Steps to Solve
- Sum the first 57 natural numbers
To find the sum of the first $n$ natural numbers, we use the formula:
$$ S_n = \frac{n(n + 1)}{2} $$
Here, $n = 57$. Plugging in the value:
$$ S_{57} = \frac{57(57 + 1)}{2} = \frac{57 \times 58}{2} $$
- Calculate the sum
Calculating the sum, we have:
$$ S_{57} = \frac{57 \times 58}{2} = \frac{3306}{2} = 1653 $$
- Calculate the average
Now, to find the average, we divide the sum by the total number of natural numbers, which is 57:
$$ \text{Average} = \frac{S_{57}}{57} = \frac{1653}{57} $$
- Final calculation for the average
Performing the division:
$$ \text{Average} = 29 $$
The average value of the first 57 natural numbers is $29$.
More Information
The average of the first $n$ natural numbers can be quickly calculated using the formula for the sum, which streamlines the process instead of adding each number individually.
Tips
- Forgetting the formula: Sometimes students might forget to use the sum formula for natural numbers and instead try to add them manually, which is more time-consuming and prone to error.
- Division mistakes: Ensure accurate division; small errors can change the entire answer when calculating the average.
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