Question

What is the average of the inclusive integers from 999 through to \(1,245 ?\)

Short answer

The average of the integers from 999 to 1245 is 1125.

Step-by-step solution

Understand the Range

Identify the first and last numbers in the range. The first number is 999 and the last number is 1245.

Determine the Total Number of Terms

The total number of terms in an inclusive range can be found using the formula \[ n = (last \ number - first \ number) + 1 \] Substituting the given values: \[ n = (1245 - 999) + 1 = 247 \]

Calculate the Sum of the Terms

Use the formula for the sum of an arithmetic series: \[ S = n \times \frac{(first \ number + last \ number)}{2} \] Substituting our values: \[ S = 247 \times \frac{(999 + 1245)}{2} = 247 \times 1122 = 277,734 \]

Find the Average

The average is given by dividing the sum of the terms by the number of terms: \[ Average = \frac{Sum \ of \ the \ terms}{Total \ number \ of \ terms} = \frac{277,734}{247} = 1,125 \]

Key concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. These differences are called the 'common difference.' In the problem above, the numbers 999 to 1245 form an arithmetic series.
To find the total number of terms in an arithmetic series, we use the formula \[ n = (last \,number - first \,number) + 1 \] where 'n' represents the total number of terms, the 'last number' is the end number in the series, and the 'first number' is the beginning number. This concept is essential for assessing the length of any series or range of integers.
We can also calculate the sum of an arithmetic series using: \[ S = n \, \frac{(first \ number + last \ number)}{2} \] Having these formulas in hand simplifies finding the sum and other properties of arithmetic series.

Average Calculation

Finding the average of a set of numbers is another common task in arithmetic. The average is the sum of all terms divided by the total number of terms. For the inclusive integers between 999 and 1245, this involves first calculating the sum of the series and then dividing that sum by the count of numbers.
In our example, after finding the sum of the series to be 277,734, we obtain the average by dividing by the total number of terms, which was 247. This gives us: \[ Average = \frac{277,734}{247} = 1,125 \] The average provides a central value that typifies a set of numbers. You often encounter such calculations in statistics and everyday problem-solving, highlighting its practical importance.

Inclusive Integers

The term 'inclusive integers' means you are considering all numbers from the starting point to the ending point, including both endpoints themselves. In this case, 999 to 1245 includes both 999 and 1245 in our calculations.
This concept is pivotal in determining the actual number of terms in a range. By treating the endpoints as included, the formula for the total number becomes \[ n = (last \,number - first \,number) + 1 \] Ignoring the '+1' would lead to an underestimation of the count. Recognizing inclusive ranges ensures you correctly account for every term in your set.

Problem-Solving Strategies

Tackling arithmetic problems effectively often involves breaking them down into smaller, manageable steps. For example, breaking down our task into identifying the range, calculating the total number of terms, finding the sum of terms, and then computing the average helped us solve the problem.
Structured problem-solving ensures you cover all necessary components and reduces the chances of errors. Here are some tips:

• Understand the problem scope – Identify start and end points.
• Break down the task – Manageable steps simplify complex problems.
• Double-check your formulas – Ensure correct application.
• Verify your results – Cross-check summations and calculations.

These strategies empower you to handle any arithmetic problem with confidence and precision.