Find the sum of all even numbers from 0 to 420.

To find the sum of all even numbers from 0 to 420, we can use the formula for the sum of an arithmetic series:

\[
S_n = \frac{n}{2} \times (a_1 + a_n)
\]

where:
- \( S_n \) is the sum of the series,
- \( n \) is the number of terms,
- \( a_1 \) is the first term, and
- \( a_n \) is the last term.

### Step 1: Identify the series components
- The first even number in the series is \( a_1 = 0 \).
- The last even number in the series is \( a_n = 420 \).
- The common difference \( d \) between consecutive even numbers is 2.

### Step 2: Determine the number of terms (\( n \))
The number of terms in an arithmetic sequence can be found using the formula:

\[
n = \frac{a_n - a_1}{d} + 1
\]

Substituting the values:

\[
n = \frac{420 - 0}{2} + 1 = \frac{420}{2} + 1 = 210 + 1 = 211
\]

So, there are 211 terms in the series.

### Step 3: Calculate the sum
Now, use the sum formula:

\[
S_{211} = \frac{211}{2} \times (0 + 420) = \frac{211}{2} \times 420 = 211 \times 210 = 44,310
\]

Thus, the sum of all even numbers from 0 to 420 is **44,310**.

Would you like to know more details or have any questions?

Here are 5 related questions you might find interesting:
1. How would you find the sum of all odd numbers between 0 and 420?
2. What is the sum of all multiples of 5 between 0 and 420?
3. How do you calculate the average of all even numbers from 0 to 420?
4. What if the series started at 2 instead of 0? How would the sum change?
5. How many even numbers are there between 100 and 420, and what is their sum?

**Tip:** When dealing with arithmetic sequences, knowing the first term, last term, and the common difference makes it easy to find the sum of the entire sequence.

Discover how to find the sum of all even numbers from 0 to 420 using the arithmetic sequence formula. This step-by-step solution explains how to calculate the sum of even numbers, perfect for students in Grades 6-8 learning about sequences and series.

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