Problem 41 - Algebra and Trigonometry

Question

Find the sum of the first 60 positive even integers.

Step-by-Step Solution

Answer

The sum of the first 60 positive even integers is 3600.

Step 1: Identify the series and it's properties

The first 60 positive even integers form an arithmetic series, where the first term \(a_1 = 2\) (the first positive even integer) and the common difference \(d = 2\). The task is to find the sum of the first 60 terms of this series.

Step 2: Find the value of the last term

As it is an arithmetic sequence, the nth term \(a_n\) of the sequence can be found using the formula \(a_n = a_1 + (n - 1) * d\). Here, \(n = 60\) hence, \(a_{60} = 2 + (60 - 1) * 2 = 120\).

Step 3: Apply the sum formula and solve

Now that we have values for \(a_1\), \(a_n\), and \(n\), we can substitute these into the sum formula. The sum \(S\) of the first 60 positive even integers is given by \(S = \frac{n}{2}(a_1 + a_n) = \frac{60}{2}(2 + 120) = 30 * 122 =3600\).