Sum of Multiples of 3 from 10 to 40: An In-Depth Guide to Arithmetic Series
Understanding the sum of multiples of a specific number, such as 3, within a given range is a fundamental skill in mathematics. In this article, we will explore how to calculate the sum of the multiples of 3 from 10 to 40, and we will introduce the concept of arithmetic series and its practical applications. By the end of this article, you will be able to solve similar problems with ease.
Introduction to Arithmetic Series
Arithmetic series are a sequence of numbers where each term is obtained by adding a constant value (known as the common difference) to the previous term. For example, the multiples of 3 form an arithmetic series: 12, 15, 18, 21, 24, 27, 30, 33, 36, 39.
In this section, we will learn about the general formula for the sum of an arithmetic series and how it can be applied to solve our specific problem.
Understanding the Problem
The numbers from 10 to 40 contain a series of multiples of 3: 12, 15, 18, 21, 24, 27, 30, 33, 36, and 39. Our goal is to calculate the sum of these numbers.
The Concept and Formula
The sum of an arithmetic series can be found using the formula: [ S frac{n}{2} times (a l) ] where ( S ) is the sum, ( n ) is the number of terms, ( a ) is the first term, and ( l ) is the last term.
In our case, the first term (( a )) is 12, the last term (( l )) is 39, and the number of terms (( n )) is 10. Substituting these values into the formula, we get:
S frac{10}{2} times (12 39)
[ S 5 times 51 255 ]
Alternative Method: Formula for Sum of an Arithmetic Sequence
Another way to find the sum of an arithmetic sequence is through the formula: [ S frac{n}{2} times [2a (n-1)d] ] where ( d ) is the common difference (in this case, ( d 3 )).
Substituting the values, we get:
S frac{10}{2} times [2 times 12 (10-1) times 3]
[ S 5 times [24 27] 5 times 51 255 ]
Further Explorations and Applications
The concept of arithmetic series can be applied to various fields, including finance, physics, and engineering. For example, in finance, it can help estimate the total interest accumulated on an investment with a fixed interest rate over a period. In physics, it can be used to calculate the total displacement or velocity changes over time.
Example of an Alternative Solution
In the given examples, there is an additional calculation involving the removal of two numbers from the series. Let's break it down and see how it affects the sum:
Problem with Specific Outliers
The given problem suggests removing the numbers 12 and 48 from the sum. However, since 48 is not between 12 and 48, we only need to remove 12 from the series. The new series will be: 15, 18, 21, 24, 27, 30, 33, 36, and 39.
The number of terms in the new series is 9. Using the formula, we get:
S frac{9}{2} times [2 times 15 (9-1) times 3]
[ S frac{9}{2} times [30 24] frac{9}{2} times 54 243 ]
To include the sum of the new series and verify its correctness, we can use the shortcut method:
S 3 times [15 times 16 / 2 - 4 times 5 / 2] 3 times [120 - 10] 3 times 110 330
This method involves the sum of consecutive natural numbers and can be useful in simplifying complex calculations.
Conclusion
Understanding how to calculate the sum of multiples of a number is crucial for solving various mathematical problems. By applying the formula for the sum of an arithmetic series, you can solve similar problems with ease. Whether you are a student, a teacher, or a professional in a related field, mastering these concepts will undoubtedly enhance your problem-solving skills. Experiment with different sequences and numbers to deepen your understanding and apply these techniques in real-world scenarios.