Question

What is the sum of all the numbers from 18 through to 40?

Short answer

667

Step-by-step solution

- Identify the first term and the last term

The first term (a) is 18 and the last term (l) is 40.

- Count the number of terms

Calculate the number of terms (n) in the sequence by using the formula: \[ n = (l - a) + 1 \]. Substitute the values: \[ n = (40 - 18) + 1 = 23 \].

- Calculate the sum of the arithmetic series

Use the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a + l) \]. Substitute the values: \[ S_{23} = \frac{23}{2} (18 + 40) = \frac{23}{2} \times 58 = 23 \times 29 = 667 \].

Key concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. For example, in the sequence 2, 5, 8, 11, the common difference is 3. The pattern repeats itself for each term.
When dealing with arithmetic series, it's important to understand the key elements: the first term (denoted as \( a \)), the last term (denoted as \( l \)), and the common difference (denoted as \( d \)). Once these elements are known, you can calculate other properties of the series, such as the number of terms and the sum.

Sum of Numbers

Calculating the sum of an arithmetic series can be done easily using a specific formula. The sum of the first \( n \) terms of an arithmetic series is given by:


This formula makes it straightforward to find the sum without needing to add each number individually.

Sequence Calculation

To solve problems involving arithmetic sequences, follow a few simple steps. First, identify the first term (\( a \)) and the last term (\( l \)).
Next, calculate the number of terms (\( n \)) in the sequence using the formula:

For the given exercise, substituting the values gives us:

Finally, use the sum formula for the arithmetic series:
Remember, breaking down the sequence calculation into these manageable steps helps you keep track and ensures accuracy.

Mathematical Problem-Solving

Solving mathematical problems like finding the sum of numbers in an arithmetic series requires a structured approach. Follow these general tips:

• Understand the problem - Read the problem statement carefully.
• Identify Given and Required - List down known values and what needs to be found.
• Use Formulas - Apply the relevant formulas to calculate the required values.
• Verify Your Solution - Double-check calculations to ensure accuracy.

By applying these steps, you can effectively approach and solve various mathematical problems, making the process more systematic and less daunting.
Remember to practice regularly to gain confidence and mastery in these solving techniques.