Write an arithmetic formula, if possible, to describe the sequence of numbers and determine \[ \begin{array}{l}a_{8} \\ a_{n}=5+7(n-1) ; a_{8}=54 \\ a_{n}=7-5(n-1) ; a_{8}=-28 \\ a_{n}=7+5(n-1) ; a_{8}=42 \\ a_{n}=5-7(n-1) ; a_{8}=-44\end{array} \]

To solve this, let's break down the formulas provided and find relevant patterns: The first formula \( a_{n}=5+7(n-1) \) can be rewritten as \( a_{n}=7n - 2 \). When we substitute \( n=8 \), \( a_{8}=7(8) - 2=54 \), which holds true! The second formula \( a_{n}=7-5(n-1) \) simplifies to \( a_{n}=-5n + 12 \). Thus, for \( n=8 \), \( a_{8}=-5(8) + 12=-28 \), also true! The third formula \( a_{n}=7+5(n-1) \) rewrites to \( a_{n}=5n + 2 \). Plugging in \( n=8 \) gives \( a_{8}=5(8) + 2=42 \), correct as well! The fourth formula \( a_{n}=5-7(n-1) \) converts to \( a_{n}=-7n + 12 \). In this case, substituting \( n=8 \) results in \( a_{8}=-7(8) + 12=-44 \), which is accurate too! So, to summarize, each formula accurately calculates the 8th term of the sequence, showcasing different linear relationships that produce varying outputs. Each formula highlights how linear sequences can evolve based on coefficients and constants. As for real-world applications, arithmetic sequences play a crucial role in finance, like calculating loan payments or savings growth. Understanding these formulas helps create budgets or forecast expenses over time efficiently. Another fun fact? Many video games use arithmetic sequences when leveling up! For instance, gaining experience points for each level often follows a sequence, helping players gauge their progress or set goals. So, whether you're tackling a math problem or gaming, understanding these sequences can be quite handy!