Exploring the Sum of the First Ten Terms of an Arithmetic Series

Understanding the sum of an arithmetic series, particularly the first ten terms, is a fundamental concept in mathematics. Let's delve into the process of calculating the sum of the series 3, 6, 9... by leveraging key formulas and concepts. This article will guide you through the methodical steps required, ensuring that the explanation is clear and comprehensive.

Understanding the Series: An Introduction to Arithmetic Sequences

The given sequence 3, 6, 9... is an example of an arithmetic series. An arithmetic series is a sequence of numbers such that the difference between any two successive members is constant. In our case, this constant difference, or common difference, is 3. This property is crucial as it simplifies the process of finding both the terms and the sum of the series.

Calculating the Sum of the First Ten Terms Using the Formula

To find the sum of the first ten terms of an arithmetic series, we can use the formula for the sum of the first n terms of an arithmetic sequence:

Sn (n/2)(2a1 (n-1)d)

where:

Sn is the sum of the first n terms. a1 is the first term of the sequence. n is the number of terms. d is the common difference.

Step-by-Step Calculation

Let's apply this formula to our sequence. We have:

a1 3 n 10 d 3

Inserting these values into the formula:

S10 (10/2)(2 * 3 (10-1) * 3)

Simplifying:

S10 5(6 27) 5 * 33 165

Alternative Method: Using the Sum Formula Directly

Alternatively, we can use the formula that directly calculates the sum of the first n terms of an arithmetic series:

Sn n/2 (a1 an)

where an is the nth term of the sequence, calculated using the formula for the nth term of an arithmetic sequence:

an a1 (n-1)d

For our sequence, to find the 10th term:

a10 3 (10-1) * 3 3 27 30

Substituting into the sum formula:

S10 10/2(3 30) 5 * 33 165

Conclusion

Through both the direct application of the sum formula and the step-by-step breakdown using the nth term formula, we have calculated the sum of the first ten terms of the arithmetic series 3, 6, 9... to be 165. This example highlights the power of mathematical formulas in simplifying complex calculations and demonstrates the importance of understanding the fundamental concepts of arithmetic sequences.

Remember, mastering the sum of arithmetic sequences can greatly enhance your problem-solving abilities in mathematics. With practice, you can confidently tackle more complex series and problems. Good luck in your studies!