Finding the nth Term of an Arithmetic Sequence

When dealing with arithmetic sequences, understanding how to find a specific term, such as the nth term, is crucial. In this article, we delve into the process of finding the term at which the number becomes 23 in the sequence 3, 5, 7, 9, and more generally, how to approach similar problems.

Understanding the Sequence

The sequence you provided, 3, 5, 7, 9, is an example of an arithmetic sequence. In an arithmetic sequence, each term after the first is obtained by adding a constant value, known as the common difference, to the previous term.

Identifying Key Parameters

For the given sequence:

The first term, (a), is 3. The common difference, (d), is 2.

General Formula for the nth Term

The general formula for the nth term of an arithmetic sequence is:

$$a_n a(n-1)d$$

This formula can be broken down as follows:

(a_n) (a) (n-1) (d)

Here, (a_n) is the nth term, (a) is the first term, (n) is the term number, and (d) is the common difference.

Step-by-Step Solution

Given the problem, we need to find the term (n) such that the term value is 23. Using the formula, we can solve for (n).

Substitute the known values into the formula: $$23 3 (n - 1) cdot 2$$ Simplify and solve for (n): $$23 3 2(n - 1)$$ $$23 - 3 2(n - 1)$$ $$20 2(n - 1)$$ $$10 n - 1$$ $$n 11$$

Thus, the term will be 23 at the 11th term of the sequence.

General Problem Interpretation

The process can be summarized with a more general approach:

Given: First term (a), common difference (d), and desired term value (a_n) Formula: ($a_n a (n - 1)d$$ Solve for (n):

Let's break down the equation to solve for (n):

$$a_n - a (n - 1)d$$ $$frac{a_n - a}{d} n - 1$$ $$n - 1 frac{a_n - a}{d}$$ $$n frac{a_n - a}{d} 1$$

Application and Flexibility

This method can be applied to any arithmetic sequence. For instance:

If (a_1 3), (d 2), and (a_n 23), then using the formula: $$23 3 (n - 1) cdot 2$$ $$20 2(n - 1)$$ $$10 n - 1$$ $$n 11$$

Similarly, for another sequence, you can follow the same steps to find the desired term number.

Conclusion

Understanding arithmetic sequences and how to find the nth term is a fundamental concept in mathematics. By breaking down the problem and using the general formula, you can efficiently solve for any term in an arithmetic sequence.

Remember, the key parameters (a), (d), and (a_n) are essential in finding the term number (n). Applying the step-by-step approach will help you solve a wide range of problems involving arithmetic sequences.