Finding the 10th Term in an Arithmetic Sequence

Arithmetic sequences are a fundamental concept in mathematics, used in various fields from finance to engineering. Understanding how to find the terms in an arithmetic sequence is crucial. Let's explore how to find the 10th term given that the first term, (a_1), is -25 and the common difference, (d), is 3.

The Formula for the Nth Term in an Arithmetic Sequence

The general formula for the (n)-th term of an arithmetic sequence is given by:

General Formula: (a_n a_1 (n - 1) cdot d)

(a_n) is the (n)-th term in the sequence. (a_1) is the first term in the sequence. (d) is the common difference, the amount by which each term increases or decreases. (n) is the term number, which is the position of the term in the sequence.

Step-by-Step Solution

Given the problem, we are asked to find the 10th term, (a_{10}), with the following values:

(a_1) -25 (d) 3 (n) 10

We will use the formula to substitute these values and solve for (a_{10}).

Start with the general formula: (a_n a_1 (n - 1) cdot d). Substitute the given values into the formula: (a_{10} -25 (10 - 1) cdot 3) Simplify inside the parentheses: (a_{10} -25 9 cdot 3) Perform the multiplication: (a_{10} -25 27) Add the values: (a_{10} 2)

Thus, the 10th term in the arithmetic sequence is 2.

Understanding the Solution

By using the formula, we can determine the value of any term in an arithmetic sequence as long as we know the first term ((a_1)) and the common difference ((d)). In this case, since the common difference is positive, the sequence increases by 3 each time. The first term is -25, and after nine steps, the value is increased by 27, resulting in the 10th term being 2.

Conclusion

In conclusion, to find the 10th term in an arithmetic sequence with a first term of -25 and a common difference of 3, we use the formula (a_{10} a_1 (10 - 1) cdot d). After substitution and simplification, we find that the 10th term is 2. Understanding and applying this formula is essential for dealing with problems related to arithmetic sequences, making it a valuable tool in various mathematical and real-world applications.

Frequently Asked Questions

What is an arithmetic sequence? An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a constant (the common difference). How do I find the common difference? The common difference ((d)) can be found by subtracting the first term from the second term: (d a_2 - a_1). How do I use the formula to find any term in the sequence? Use the formula (a_n a_1 (n - 1) cdot d). Substitute the known values and solve for the desired term.

If you have more questions or need further assistance, feel free to ask!