Find the sum of the first 60 positive even integers.

Verified step by step guidance

1. Recognize that the sequence of the first 60 positive even integers is an arithmetic sequence where the first term \( a_1 = 2 \) and the common difference \( d = 2 \).
2. 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, \( a_1 \) is the first term, and \( a_n \) is the last term.
3. Determine the last term \( a_n \) of the sequence. Since the sequence is even integers, \( a_n = 2n \). For 60 terms, \( a_{60} = 2 \times 60 \).
4. Substitute \( n = 60 \), \( a_1 = 2 \), and \( a_{60} = 120 \) into the sum formula: \( S_{60} = \frac{60}{2} (2 + 120) \).
5. Simplify the expression to find the sum: \( S_{60} = 30 \times 122 \).

Key Concepts

Even Integers

Even integers are whole numbers that are divisible by 2 without a remainder. They can be expressed in the form of 2n, where n is an integer. The sequence of positive even integers starts from 2 and continues as 2, 4, 6, 8, and so on. Understanding this concept is crucial for identifying the specific integers involved in the problem.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term is obtained by adding a constant difference to the previous term. The sum of the first n terms can be calculated using the formula S_n = n/2 * (a + l), where S_n is the sum, n is the number of terms, a is the first term, and l is the last term. This concept is essential for efficiently calculating the sum of the first 60 positive even integers.

Formula for the Sum of Even Integers

The sum of the first n positive even integers can be calculated using the formula S = n(n + 1), where n is the number of even integers. This formula arises from the properties of arithmetic series and simplifies the calculation process. For the question at hand, applying this formula will yield the sum of the first 60 positive even integers directly.