Understanding Different Types of Sequences

1. Arithmetic Sequences:

   • What It Is: A list of numbers where you add the same amount each time.
   • Formula: You can find the nth term like this: $a_n = a_1 + (n-1)d$.
   • Example: The numbers $2, 5, 8, 11$ (Here, you add $3$ each time).

2. Geometric Sequences:

   • What It Is: A list of numbers where you multiply by the same amount each time.
   • Formula: You can find the nth term like this: $a_n = a_1 \cdot r^{(n-1)}$.
   • Example: The numbers $3, 6, 12, 24$ (Here, you multiply by $2$ each time).

3. Harmonic Sequences:

   • What It Is: A list of numbers that are the upside-down (reciprocal) of an arithmetic sequence.
   • Example: The numbers $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}$.

4. Fibonacci Sequences:

   • What It Is: A list of numbers where each number is the sum of the two numbers before it.
   • Formula: You can find the nth term like this: $F_n = F_{n-1} + F_{n-2}$.
   • Example: The numbers $0, 1, 1, 2, 3, 5, 8$.

Understanding these different types of sequences is important for solving problems in pre-calculus.