Problem 43 - Algebra and Trigonometry

Question

Find the sum of the even integers between 21 and 45

Step-by-Step Solution

Answer

The sum of the even integers between 21 and 45 is 396

Step 1: Find the first and last even numbers within the range

The first step is to find the smallest even number greater than 21 and the largest even number less than 45. The first even number in this range would be 22 and the last would be 44

Step 2: Calculate number of terms

Next, find the number of even numbers within this range. Since this sequence of even integers has a common difference of 2, the number of terms, \(n\), can be calculated as \((l - a)/2 + 1 \), where \(a\) is the first term and \(l\) is the last term. So, \(n = (44-22)/2 + 1 = 12\)

Step 3: Apply the sum formula

Now, apply the formula for the arithmetic series sum, \(S_n = n/2 * (a+l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term. Substituting values into the formula gives \(S_n = 12/2 * (22 + 44) = 6 * 66 = 396\)