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An arithmetic sequence is a series of numbers in which each term after the first is formed by adding a constant difference to the preceding term. Here's a simple way to understand it. Imagine a number line: you start at a number, say 2, and then keep adding the same number, like 2, to each result. That's an arithmetic sequence!

In the exercise, the even numbers from 1 to 100 form an arithmetic sequence. The first term is 2, and the constant difference is also 2, which means we add 2 to get from one term to the next (e.g., 2, 4, 6, and so on).

• The sequence can be represented by the formula for the nth term: \( a_n = 2n \, \), where \( n \, \) is the position of the term in the sequence.
• For the sequence from 2 to 100, the nth term formula helps us calculate that there are 50 terms.

Understanding this basic concept makes it clear how to figure out the pattern and the number of terms needed for summation.

Even numbers are integers that can be divided by 2 without leaving a remainder. These numbers play a neat role when working within arithmetic sequences because they naturally form a sequence with a consistent difference of 2 between consecutive terms.

For example, the numbers 2, 4, 6, 8, are all even and are part of this predictable sequence. In our target range of 1 to 100:

• Even numbers are obtained by multiplying 2 by each integer from 1 to 50, so you get 2, 4, 6,..., 100.
• The smallest even number is 2 and the largest in our range is 100.

This predictability makes it simple to identify the terms we need when working out a sum or checking them using a graphing calculator.

The summation formula for arithmetic sequences is a powerful tool for finding the total of all numbers in the sequence, without adding them one by one. Imagine having to add each of the 50 even numbers from 2 to 100 by hand – it could be quite tasking! That's where the formula comes in handy.

The formula to find the sum of an arithmetic sequence is \( S_n = \frac{n}{2} (a_1 + a_n) \). Here's how to use it:

• \( n \, \) is the number of terms. In our exercise, there are 50 terms.
• \( a_1 \, \) is the first term, which is 2.
• \( a_n \, \) is the last term, which is 100.

To find the sum (\( S_{50} \)), simply plug in these numbers: \( S_{50} = \frac{50}{2} (2 + 100) = 25 imes 102 = 2550 \).

This formula saves time, ensuring you get the correct total efficiently. Using a graphing calculator to verify your work can also bolster confidence in your answer.