Solving for the Common Difference in an Arithmetic Sequence

Arithmetic sequences are a fundamental concept in mathematics, commonly used in various fields such as finance, physics, and computer science. One of the key features of an arithmetic sequence is the common difference, denoted as d, which represents the constant difference between consecutive terms. This article demonstrates how to find the common difference of a sequence given specific conditions on the terms.

Introduction to the Problem

The problem at hand involves an arithmetic sequence where 10 times the 10th term of the sequence is equal to 15 times the 15th term. Given that the first term of the sequence is 48, we will use the general formula for the nth term of an arithmetic sequence to solve for the common difference d.

Using the Nth Term Formula

The nth term of an arithmetic sequence can be expressed as:

a_n a (n-1)d

Where:

a is the first term of the sequence. d is the common difference. n is the term number.

Given that a 48, we can represent the 10th and 15th terms as follows:

a_{10} 48 9d

a_{15} 48 14d

Solving the Problem

We are given that 10 times the 10th term is equal to 15 times the 15th term. This can be mathematically represented as:

10a_{10} 15a_{15}

Substituting the expressions for a_{10} and a_{15} into this equation:

10(48 9d) 15(48 14d)

Expanding both sides, we get:

480 90d 720 210d

Rearranging the equation to isolate d:

480 - 720 210d - 90d

-240 120d

Solving for d gives:

d frac{-240}{120} -2

Thus, the common difference d is boxed{-2}.

Conclusion

The common difference of the arithmetic sequence is determined to be -2 based on the given conditions. Understanding and solving such problems is crucial for mastering the concepts of arithmetic sequences, which have numerous real-world applications.