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The n-th term formula is a fundamental concept in arithmetic sequences. It allows us to find any term in the sequence based on the first term and the common difference. This formula is expressed as:\[ a_n = a_1 + (n-1) \, d \]Where:

• \( a_n \) is the n-th term we are looking for.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the position of the term in the sequence.
• \( d \) is the common difference between consecutive terms.

This equation helps us calculate the value of any term given the position \( n \). It is essential in solving problems where only a few terms of the sequence are known.

In our example, knowing two terms from the sequence gives us a system of equations. Here's what happened:First, we applied the n-th term formula to each of the given terms:

• For \( a_{12} = 60 \), we wrote: \( a_1 + 11d = 60 \).
• For \( a_{20} = 84 \), we wrote: \( a_1 + 19d = 84 \).

These two expressions form our system of equations. Solving these equations simultaneously helps us determine unknowns like the first term \( a_1 \) and the common difference \( d \).By expressing known information in this way, we can systematically solve for the variables. Here, arithmetic sequence problems often rely on this approach to link different pieces of information together.

The common difference \( d \) is a key element of an arithmetic sequence. It defines how much each term increases (or decreases) from the previous term. Here's the importance of the common difference:- It remains constant throughout the entire sequence.- In our sequence problem, we found \( d \) by solving the system of equations.After setting up our system of equations, we could subtract:\\((a_1 + 19d) - (a_1 + 11d) = 84 - 60\)Simplifying this gives us: \[ 8d = 24 \]Dividing through by 8, we find that \( d = 3 \).Understanding the common difference helps predict any term's value without listing every term in the sequence.

Solving an arithmetic sequence problem involves a series of well-defined steps. Let's break down the process:1. **Identify the Formula:** Start by understanding and writing down the n-th term formula: \( a_n = a_1 + (n-1) \, d \). Recognize the terms you need to find.2. **Express Known Terms:** Use the formula on each known term to create equations. For instance, for \( a_{12} = 60 \), it became \( a_1 + 11d = 60 \).3. **Set Up Equations:** Translate the expressions into a solvable system of equations.4. **Solve for Variables:** Subtract equations or use substitution to solve for one of the variables, typically starting with \( d \). In this example, we subtracted to find \( d = 3 \).5. **Backsolve for Remaining Terms:** Use the value found (\( d = 3 \)) to find other unknowns like \( a_1 \).By following these steps carefully, you will be able to solve any arithmetic sequence problem systematically and accurately.