Even Numbers Introduction

An even number is any integer that is divisible by 2 without leaving a remainder. In simpler terms, an even number can be expressed in the form \(2n\), where \(n\) is an integer. Examples of even numbers include 2, 4, 6, 8, and so on.

Identifying Even Numbers from 1 to 40

To find the sum of even numbers from 1 to 40, we first need to list all the even numbers within this range. The even numbers start from 2 and go up to 40, including both.

• The first even number in the range is 2.
• The last even number in the range is 40.
• The sequence of even numbers is: 2, 4, 6, ..., 40.

This sequence forms an arithmetic progression (AP) because the difference between consecutive terms is constant. In this case, the common difference \(d\) is \(4 - 2 = 2\).

Counting Even Numbers (Number of Terms)

To calculate the sum of an arithmetic progression, we need to know the number of terms (\(n\)) in the sequence. For the even numbers from 1 to 40, we can find \(n\) in a few ways:

1. Using the formula for the nth term of an AP:
   The formula is \(a_n = a_1 + (n-1)d\), where:
   • \(a_n\) is the last term = 40
   • \(a_1\) is the first term = 2
   • \(d\) is the common difference = 2
   • \(n\) is the number of terms
   Substituting the values into the formula: $$40 = 2 + (n-1)2$$ Subtract 2 from both sides: $$38 = (n-1)2$$ Divide by 2: $$\frac{38}{2} = n-1$$ $$19 = n-1$$ Add 1 to both sides: $$n = 19 + 1$$ $$n = 20$$ So, there are 20 even numbers from 1 to 40.
2. Simple division:
   Since every second number is even, and the sequence starts from 2, we can simply divide the last even number by 2 to find the count of even numbers up to that point. $$n = \frac{\text{Last even number}}{2} = \frac{40}{2} = 20$$ This method works efficiently when the sequence starts from 2.

Sum Calculation of Even Numbers

Now that we know the first term (\(a_1 = 2\)), the last term (\(a_n = 40\)), and the number of terms (\(n = 20\)), we can use the sum formula for an arithmetic progression:

The sum \(S_n\) of an arithmetic progression is given by: $$S_n = \frac{n}{2}(a_1 + a_n)$$ Substituting the values:

Therefore, the sum of even numbers from 1 to 40 is 420.

Final Answer Verification

Based on our detailed calculation using the principles of arithmetic progression, the sum of even numbers from 1 to 40 is 420. This matches one of the provided options, confirming the accuracy of our steps.