Question

REASONING Find the sum of the positive odd integers less than 300 . Explain your reasoning.

Short answer

The sum of the positive odd integers less than 300 is 22500.

Step-by-step solution

Identify the sequence

Firstly, the sequence needs to be identified. We're dealing with odd numbers, which could be represented as 1, 3, 5, 7, etc.

Find the common difference

In this arithmetic sequence, the common difference\(d\) is 2, as each succeeding term is 2 more than the preceding term.

Calculate the number of terms

The last term\(l\) in the sequence less than 300 is 299, and the first term\(a\) is 1. The formula to calculate the number of terms \(n\) in an arithmetic series is \((l - a) / d + 1\). So plug the values into equation, \((299 - 1) / 2 + 1 = 150\). Hence, there are 150 terms.

Calculate the sum

To find the sum \(S\) of an arithmetic series, the formula \(n/2 * (a + l)\) is used. Here, \(n = 150\), \(a = 1\) and \(l = 299\). Substitute these into the formula, we get \(150/2 * (1 + 299) = 22500\)

Key concepts

Odd Numbers

Odd numbers are integers that are not divisible by 2. This means when you divide an odd number by 2, you'll get a remainder of 1. Examples include 1, 3, 5, 7, 9, and so on.
Unlike even numbers, which alternate with odd numbers on the number line, odd numbers are critical in various mathematical patterns and sequences.
They are regularly spaced by a difference of 2, which makes them an interesting subject for arithmetic sequences.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is the sum of the previous term and a constant, called the "common difference."
For example, in the sequence of odd numbers 1, 3, 5, 7, each number is obtained by adding 2 to the previous number.
This helps determine not just the possible numbers in the sequence, but also how many such numbers can be included in a given range.

Common Difference

In an arithmetic sequence, the common difference is the consistent amount that separates each pair of successive terms.
For the odd numbers sequence, the common difference is 2 since each odd number is two more than the last.
Understanding the common difference allows us to predict future numbers in the sequence and calculate various properties, such as the total number of terms or their sum.

Sequence Reasoning

Sequence reasoning involves understanding the pattern and rules that define a sequence, which allows us to make predictions or solve problems involving sequences.
For instance, by recognizing the pattern in the odd numbers up to 299, we can use the formula for the sum of an arithmetic series to find the total of these numbers.
In this exercise, knowing that the series starts at 1 and ends at 299, with a common difference of 2, allows us to calculate quickly and accurately using established mathematical techniques.