The bids in an online auction are represented by the arithmetic sequence shown below. Write an explicit formula to represent the bids as an arithmetic sequence. What is the twelfth bid? \( 214,225,236,247, \ldots \) \( a_{n}=\square \) (Type an expression using \( n \) for the variable.)

To derive the explicit formula for the arithmetic sequence of bids, we can start by identifying the first term, \( a_1 = 214 \), and the common difference, \( d = 225 - 214 = 11 \). The explicit formula for an arithmetic sequence can be expressed as \( a_n = a_1 + (n - 1) \cdot d \). So, our formula becomes \( a_n = 214 + (n - 1) \cdot 11 \). To find the twelfth bid \( a_{12} \), we substitute \( n = 12 \) into our formula: \[ a_{12} = 214 + (12 - 1) \cdot 11 = 214 + 121 = 335. \] Thus, the twelfth bid is \( 335 \). Now, the explicit formula for the bids is \( a_n = 214 + 11(n - 1) \). How fun is it to note that auctions have been around since ancient Rome, where not just goods but also slaves and real estate were bid on! Fast forward to today, and you can snag everything from a rare comic book to a vintage car with just a click! If you're venturing into online auctions, remember that timing can be everything! Many bidders wait until the last moments to submit their highest offers, so don't get too comfortable too early. Common mistakes include not reading item descriptions carefully or forgetting to factor in shipping costs, which can turn a "great deal" into a budget buster!