Arithmetic sequences are important in many real-life situations. That's why they're a key topic in Grade 12 Pre-Calculus.

To understand these sequences, students need to know two main formulas:

1. Explicit Formula: This is given by ( a_n = a_1 + (n - 1)d ).
2. Recursive Formula: Here, it looks like this: ( a_n = a_{n-1} + d ).

These formulas help us find patterns when things grow in a straight line.

Real-World Examples:

1. Saving Money: Imagine you save $100 every month. Your savings create an arithmetic sequence where:

   • ( a_1 = 100 ) (the amount you save in the first month),
   • ( d = 100 ) (how much you add each month).

   After ( n ) months, your total savings can be found using the formula:

   • ( a_n = 100 + (n - 1) * 100 = 100n ).

2. Building Heights: Think about a building where each floor is 4 feet tall. The height after ( n ) floors follows an arithmetic sequence. In this case:

   • ( a_1 = 4 ) (the height of the first floor),
   • ( d = 4 ) (the height added for each new floor).

   So, the height after ( n ) floors is calculated with:

   • ( a_n = 4 + (n - 1) * 4 = 4n ).

By learning about arithmetic sequences, students can understand and manage real-life situations that involve patterns and making predictions!