Question Write an explicit formula for \( a_{n} \), the \( n^{\text {th }} \) term of the sequence \( 25,20,15, \ldots \) Answer Attempt 1 out of 3 \( a_{n}=\square \)

To find the explicit formula for the sequence \( 25, 20, 15, \ldots \), we can observe that this is an arithmetic sequence where each term decreases by 5. The first term \( a_1 \) is 25, and the common difference \( d \) is -5. Using the formula for the \( n^{\text{th}} \) term of an arithmetic sequence, which is \( a_n = a_1 + (n-1)d \), we can plug in our values: \[ a_n = 25 + (n-1)(-5) \] This simplifies to: \[ a_n = 25 - 5(n-1) = 30 - 5n \] So, the explicit formula is \( a_n = 30 - 5n \). Here's a fun twist: You might think of this sequence as a countdown! Imagine you’re at a party that starts at 25 balloons and every hour, five balloons float away. The formula keeps track of how many balloons you have left with each tick of time, making it a perfect playlist for a balloon-themed party! Also, you can think of linear sequences like this one as the 'infinite staircase'. Each step down (like from 25 to 20) is a consistent drop. This kind of thinking can help visualize not just the numbers, but their relationships—just like counting your steps in life and knowing every move gets you closer to your goals!