In Grade 12 Pre-Calculus, it's really important to understand the differences between explicit and recursive formulas for geometric sequences. This will help you get a better grasp of sequences and series.

Definitions:

1. Geometric Sequence: This is a list of numbers where each number after the first is found by multiplying the previous number by a fixed number. This number is known as the common ratio ($r$).

2. Explicit Formula: This formula lets you directly calculate the $n^{th}$ term of the sequence. For a geometric sequence, it looks like this: $a_n = a_1 \cdot r^{(n - 1)}$ Here’s what each part means:

   • $a_n$: the $n^{th}$ term
   • $a_1$: the first term
   • $r$: the common ratio
   • $n$: the number of the term

3. Recursive Formula: This formula defines the terms by relying on the previous term. For a geometric sequence, it looks like this: $a_n = a_{n-1} \cdot r$ And you need to know: $a_1 = a_1$ This means you start with the first term and calculate each term using the one before it.

Key Differences:

• Form of Expression:

  • Explicit Formula: You can find the $n^{th}$ term directly. It’s quick and you don’t have to worry about all the earlier terms.
  • Recursive Formula: You need the previous terms. For example, to find $a_5$, you first need $a_4$, then $a_3$, and so on.

• Usability:

  • Explicit Formula: Great for finding terms that are far away from the first term or when you have a lot of data.
  • Recursive Formula: Works well in programming or when each term builds off the last one, like in simulations.

• Initial Conditions:

  • Explicit Formula: Just needs the first term and the common ratio to work.
  • Recursive Formula: Needs the first term and the rule to find the next terms.

• Complexity:

  • Explicit Formula: Usually easier to use, especially when you're only finding one term.
  • Recursive Formula: Can get tricky for higher terms unless you have good tools to help you.

Application Example:

Let’s say we have a geometric sequence where the first term is $a_1 = 3$ and the common ratio is $r = 2$.

• Explicit Calculation:
  • To find the fifth term, we do: $a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 16 = 48$
• Recursive Calculation:
  • First term: $a_1 = 3$
  • Second term: $a_2 = a_1 \cdot 2 = 3 \cdot 2 = 6$
  • Third term: $a_3 = a_2 \cdot 2 = 6 \cdot 2 = 12$
  • Fourth term: $a_4 = a_3 \cdot 2 = 12 \cdot 2 = 24$
  • Fifth term: $a_5 = a_4 \cdot 2 = 24 \cdot 2 = 48$

By understanding these differences, you can choose the best method for your calculations in math!