Find the sum of the even integers between 21 and 45.

Verified step by step guidance

1. Identify the even integers between 21 and 45. The smallest even integer greater than 21 is 22, and the largest even integer less than 45 is 44.
2. List the even integers between 22 and 44: 22, 24, 26, ..., 44.
3. Recognize that this sequence is an arithmetic sequence where the first term \(a_1 = 22\) and the last term \(a_n = 44\), with a common difference \(d = 2\).
4. Use the formula for the sum of an arithmetic sequence: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \(n\) is the number of terms.
5. Calculate \(n\) using the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1) \cdot d\). Solve for \(n\) and then use it to find the sum \(S_n\).

Key Concepts

Even Integers

Even integers are whole numbers that are divisible by 2 without a remainder. They can be represented in the form of 2n, where n is an integer. In the context of the question, identifying the even integers between 21 and 45 is essential for calculating their sum.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference. The sum of an arithmetic series can be calculated using the formula S = n/2 * (a + l), where n is the number of terms, a is the first term, and l is the last term. This concept is useful for efficiently summing the identified even integers.

Range of Integers

The range of integers refers to the set of integers that fall between two specified values. In this case, the integers between 21 and 45 include both endpoints, but the task specifically focuses on the even integers within this range. Understanding how to identify and list these integers is crucial for solving the problem.