To get better at changing between explicit and recursive formulas for geometric sequences, it's important to know what each form means.

1. Explicit Formula: This is written as $a_n = a_1 \cdot r^{(n - 1)}$, where:

   • $a_n$ is the $n^{th}$ term, which means the term in that position.
   • $a_1$ is the first term of the sequence.
   • $r$ is the common ratio, which is how much each term is multiplied by to get the next term.

2. Recursive Formula: This formula looks like $a_n = a_{n-1} \cdot r$, with:

   • $a_1$ being the first term again.

Example:

Let’s say $a_1 = 3$ and $r = 2$:

• Explicit formula: You would write it as $a_n = 3 \cdot 2^{(n - 1)}$.
• Recursive formula: This would look like $a_n = 2 \cdot a_{n-1}$ and remember that $a_1 = 3$.

To get really comfortable with these formulas, try making your own sequences. Switch back and forth between the explicit and recursive forms. This will help you understand them better!