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In arithmetic sequences, the **common difference** is a key element that defines the uniform change from one term to the next. It's what makes the sequence "arithmetic." For any two consecutive terms \(a_{m}\) and \(a_{m+1}\), the common difference \(d\) can be found by the formula:

• \(d = a_{m+1} - a_{m}\)

This consistent "step" is crucial because it helps in forming the sequence and finding additional terms. In the given problem, the terms \(a_{15} = 8\) and \(a_{17} = 2\) allow us to compute the common difference, which turns out to be \(-3\). Knowing \(d = -3\), we understand each term decreases by 3 compared to the one before it. This information not only helps in finding previous and next terms but also assists in deriving other sequence properties.

The **n-th term formula** is a powerful equation that provides the value of any term in an arithmetic sequence. This formula is useful for constructing the sequence using the first term and common difference:

• \(a_n = a_1 + (n-1) \times d\)

In this equation:

• \(a_1\) is the first term of the sequence.
• \(n\) is the position of the term you want to find.
• \(d\) is the common difference.

For the example provided, substituting the known values gives us \(a_n = 50 + (n-1)(-3)\). Simplifying this, we get the general formula \(a_n = 53 - 3n\). This formula not only simplifies finding any particular term quickly but also reveals the linear nature of arithmetic sequences.

The **first term calculation** is often the launching pad for understanding and applying concepts related to arithmetic sequences. Knowing the first term is crucial because it provides a starting point for applying the n-th term formula to find other terms. To find \(a_1\), we leverage a known term from the sequence along with the common difference.For instance, using \(a_{15} = 8\) and the common difference \(d = -3\), we can set up the formula \(a_{15} = a_1 + 14(-3)\). Solving this:

• \(8 = a_1 - 42\)
• \(a_1 = 50\)

Hence, the first term \(a_1\) is 50. Once the first term is known, it simplifies the process of finding any subsequent or preceding term using the n-th term formula. This makes navigation through the sequence much more straightforward.