Find the sum of the first 60 positive even integers.

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Recognize that the first 60 positive even integers form an arithmetic sequence where the first term \(a_1\) is 2 and the common difference \(d\) is also 2.

Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a_1 + (n-1)d\). Here, calculate the 60th term \(a_{60}\) as \(2 + (60-1) \times 2\).

Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence: \(S_n = \frac{n}{2} (a_1 + a_n)\).

Substitute \(n=60\), \(a_1=2\), and the value of \(a_{60}\) from step 2 into the sum formula to set up the expression for the sum.

Simplify the expression to find the sum of the first 60 positive even integers.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Even Integers

Even integers are whole numbers divisible by 2 without a remainder. The first positive even integers start from 2 and increase by 2 each time (2, 4, 6, ...). Understanding this sequence helps identify the terms involved in the problem.

Arithmetic Sequence

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. Here, the even integers form an arithmetic sequence with a common difference of 2. Recognizing this allows the use of formulas to find sums efficiently.

Sum of an Arithmetic Series

The sum of the first n terms of an arithmetic sequence can be found using the formula S_n = n/2 * (first term + last term). This formula simplifies calculating the total of many terms without adding each individually.