Find the sum of the odd integers between 30 and 54.

Verified step by step guidance

1. Identify the range of odd integers between 30 and 54. Since 30 is even, the first odd integer greater than 30 is 31, and since 54 is even, the last odd integer less than 54 is 53.
2. List the odd integers between 30 and 54: 31, 33, 35, ..., 53. These form an arithmetic sequence where the first term $a=31$ and the last term $l=53$.
3. Determine the common difference $d$ of the sequence, which is 2 because odd numbers increase by 2.
4. Calculate the number of terms $n$ in the sequence using the formula for the nth term of an arithmetic sequence: $l=a+\; (n\; -\; 1)d$. Solve for $n$.
5. Use the formula for the sum of an arithmetic sequence: $S\_n\; =\; \backslash frac\{n\}\{2\}\; (a\; +\; l)$. Substitute the values of $n$, $a$, and $l$ to express the sum.

Key Concepts

Identifying Odd Integers

Odd integers are whole numbers that are not divisible by 2, typically ending with digits 1, 3, 5, 7, or 9. To find odd integers between two numbers, list all numbers in that range and select those that are odd.

Arithmetic Sequences

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. Odd integers form an arithmetic sequence with a common difference of 2, which helps in efficiently finding sums without listing all terms.

Sum of an Arithmetic Sequence

The sum of an arithmetic sequence can be found using the formula S = n/2 * (first term + last term), where n is the number of terms. This formula simplifies adding many numbers by using only the first and last terms and the count of terms.