Exploring Arithmetic Sequences: Sum of the First Three Terms of An5n-3

In mathematics, especially in sequences and series, understanding arithmetic sequences is fundamental. An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. Here, we will explore the sum of the first three terms of the arithmetic sequence defined by the formula An 5n - 3.

Understanding the Formula: An 5n - 3

The formula An 5n - 3 is a general expression for an arithmetic sequence where:

An represents the nth term of the sequence. n is the position of the term in the sequence. The common difference is 5. The first term A1 is calculated as 5*1 - 3 2.

Calculating the First Three Terms

Let's determine the first three terms of the sequence using n 0, 1, 2 and the formula An 5n - 3:

For n 0:
A0 5*0 - 3 -3 For n 1:
A1 5*1 - 3 2 For n 2:
A2 5*2 - 3 7

The Sum of the First Three Terms

To find the sum of the first three terms, we add up the values we found:

Sum A0 A1 A2 -3 2 7 6

Generalizing the Sum for Any Starting Point

The sum of the first three terms can be generalized for any starting point of n. For example, if we set n 123 in the formula An 5n - 3, we get:

A123 5*123 - 3 612 - 3 609 A124 5*124 - 3 620 - 3 617 A125 5*125 - 3 625 - 3 622

For the first three terms with n 123:
Sum A123 A124 A125 609 617 622 1848

Conclusion

Understanding arithmetic sequences is crucial for many areas of mathematics and real-world applications. By calculating the sum of the first three terms of the sequence using the formula An 5n - 3, we can gain insights into the behavior and properties of such sequences. Whether starting from n 0 or any other value, the process remains consistent, making these sequences a valuable tool for problem-solving and analysis.