To find the sum of the first 100 odd numbers, we need to understand the pattern of odd numbers and how they can be added together. In this guide, we will walk through the process step by step, providing an easy-to-follow explanation of how to solve this problem.
How to Find the Sum of the First 100 Odd Numbers
Understanding Odd Numbers and Their Pattern
Odd numbers are integers that cannot be evenly divided by 2, meaning they have a remainder of 1 when divided by 2. The first few odd numbers are 1, 3, 5, 7, 9, and so on. Notice the pattern: each odd number can be represented as 2n+1, where n is an integer starting from 0.
1 = 2(0) + 1 3 = 2(1) + 1 5 = 2(2) + 1 7 = 2(3) + 1
We can use this pattern to find the sum of the first 100 odd numbers.
Using the Arithmetic Series Formula
To find the sum of a sequence of numbers, we can use the arithmetic series formula:
Sum = (n * (a1 + an)) / 2
where n is the number of terms in the series, a1 is the first term, and an is the last term. In our case, we are looking for the sum of the first 100 odd numbers, so n = 100. We also know the first term, a1 = 1. To find the 100th odd number (an), we can use the pattern mentioned above:
an = 2(n-1) + 1
Plugging in n = 100, we get:
an = 2(100-1) + 1 = 2(99) + 1 = 198 + 1 = 199
Now we have all the values needed to find the sum using the arithmetic series formula:
Sum = (n * (a1 + an)) / 2 Sum = (100 * (1 + 199)) / 2
Calculating the Sum of the First 100 Odd Numbers
Now that we have all the information needed, we can plug in the values and solve for the sum:
Sum = (100 * (1 + 199)) / 2 Sum = (100 * 200) / 2 Sum = 20000 / 2 Sum = 10000
The sum of the first 100 odd numbers is 10,000.
Summary
In this tutorial, we learned how to find the sum of the first 100 odd numbers by following these steps:
- Understand the pattern of odd numbers, which can be represented as 2n + 1, where n is an integer.
- Use the arithmetic series formula to find the sum: Sum = (n * (a1 + an)) / 2.
- Calculate the 100th odd number using the pattern: an = 2(n-1) + 1.
- Plug in the values for n, a1, and an into the formula, and solve for the sum.
By following these steps, we found that the sum of the first 100 odd numbers is 10,000.