Understanding Arithmetic Sequences

An arithmetic sequence is a sequence of numbers where each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This type of sequence has numerous applications in various fields, from mathematics to real-world scenarios like financial forecasting or pattern recognition.

The General Formula for the nth Term

To find the (text{n})-th term of an arithmetic sequence, you can use the general formula:

(text{a}_text{n} text{a}_1 (text{n} - 1)d)

(text{a}_1) is the first term of the sequence. (d) is the common difference between consecutive terms. (text{n}) is the term number you want to find.

Example: 35th Term of the Sequence 3, 9, 15, 21,...

Let's walk through the steps to find the 35th term in the arithmetic sequence 3, 9, 15, 21...

Identify the First Term: The first term (text{a}_1 3). Determine the Common Difference: The common difference (d 9 - 3 6). Apply the General Formula:

(text{a}_text{n} text{a}_1 (text{n} - 1)d)

(text{a}_{35} 3 (35 - 1) cdot 6)

(text{a}_{35} 3 34 cdot 6)

(text{a}_{35} 3 204)

(text{a}_{35} 207)

The 35th term in the sequence is 207.

Sequence Pattern

Here is the first few terms of the sequence for reference, to illustrate the pattern:

3, 9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99, 105, 111, 117, 123, 129, 135, 141, 147, 153, 159, 165, 171, 177, 183, 189, 195, 201, 207, 213, 219, 225, 231, 237, 243, 249, 255, 261, 267, 273, 279, 285, 291, 297, 303, 309, 315, 321, 327...

Conclusion

By understanding and applying the general formula for the (text{n})-th term of an arithmetic sequence, you can efficiently calculate any term in the sequence, including the 35th term as we demonstrated with the sequence 3, 9, 15, 21. This method is a fundamental tool in arithmetic and has applications in many practical scenarios.