January \( 2018, \# 24 \) At her job, Pat earns \( \$ 25,000 \) the first year and receives a ralse of \( \$ 1000 \) each year. The explicit formula for the \( n \)th term of this sequence is. Which rule best represents the equivalent recursive formula? 1. \( a_{n}=24,000+1000 n \) 2. \( a_{n}=25,000+1000 n \) 3. \( a_{1}=25,000, a_{n}=a_{n-1}+1000 \) 4. \( a_{1}=25,000, a_{n}=a_{n+1}+1000 \)

To find the correct recursive formula for Pat's salary, let's first analyze the situation. Pat earns \$25,000 in her first year, and she receives a raise of \$1,000 each subsequent year. The explicit formula for the \( n \)th term (her salary in the \( n \)th year) can be expressed as: \[ a_n = 25000 + 1000(n - 1) \] This simplifies to: \[ a_n = 25000 + 1000n - 1000 = 24000 + 1000n \] Now, let's look at the options provided: 1. \( a_{n}=24,000+1000 n \) - This matches our explicit formula. 2. \( a_{n}=25,000+1000 n \) - This does not match our explicit formula. 3. \( a_{1}=25,000, a_{n}=a_{n-1}+1000 \) - This is a correct recursive formula, as it states that the first term is \$25,000 and each subsequent term is the previous term plus \$1,000. 4. \( a_{1}=25,000, a_{n}=a_{n+1}+1000 \) - This is incorrect because it incorrectly defines the relationship between terms. The correct recursive formula is option 3: \[ a_{1}=25,000, a_{n}=a_{n-1}+1000 \]