Problem 44 - Algebra and Trigonometry

Question

Find the sum of the odd integers between 30 and 54

Step-by-Step Solution

Answer

The sum of the odd integers between 30 and 54 is 504

Step 1: Identify the Range

First, identify the odd integers between 30 and 54. The odd integers begin from 31 as 30 is an even number. Therefore, the sequence of odd numbers starts from 31 and ends at 53.

Step 2: Determine the Range's Terms

Count the number of terms in the range. The first term \(a_1\) is 31 and the last term \(a_n\) is 53. These integers make up an arithmetic sequence where each subsequent term is found by adding 2 to the previous term. The formula for finding the number of terms \(n\) in an arithmetic sequence is \(n = \frac{((a_n - a_1)/d) + 1}\), where \(d\) is the common difference between the terms, which is 2 in this case. Substituting the given values, we find 12 terms.

Step 3: Calculate the Sum

Use the formula for the sum \(s\) of an arithmetic sequence \(s = n/2 * (a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) is the first term and \(a_n\) is the last term. Substituting the values, the sum is 504.