Sum of the First 40 Terms of an Arithmetic Sequence

In this article, we will explore how to find the sum of the first 40 terms of the arithmetic sequence 3, 6, 9, and other closely related concepts. Understanding these concepts is essential for anyone looking to improve their skills in algebra and number theory.

Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In the given sequence 3, 6, 9, ..., the first term a_1 is 3, and the common difference d is 3 (since 6 - 3 3, 9 - 6 3, and so on).

The General Formula for the nth Term

To find the last term (the 40th term) of the sequence, we use the formula for the nth term of an arithmetic sequence:

A_n A_1 (n-1)d

Plugging in the values for the 40th term:

A_40  3   (40 - 1) × 3A_40  3   39 × 3A_40  3   117A_40  120

Calculating the Sum of the First 40 Terms

The sum of the first n terms of an arithmetic sequence can be calculated using the following formula:

S_n frac{n}{2} [2a_1 (n-1)d]

For the first 40 terms, substituting the values:

S_40  frac{40}{2} [2 × 3   (40 - 1) × 3]S_40  20 [6   39 × 3]S_40  20 [6   117]S_40  20 [123]S_40  2460

Conclusion

In summary, the sum of the first 40 terms of the arithmetic sequence 3, 6, 9, ..., is 2460. The 40th term of the sequence is 120. Mastering the formulas and techniques for working with arithmetic sequences can greatly enhance your mathematical skills, providing a solid foundation for more complex problems and real-world applications.

Keywords

- Arithmetic sequence

- Sum of terms

- Common difference