To understand recursive and explicit formulas in sequences, let’s break it down:

Recursive Formulas

• These formulas use one or more previous numbers in the sequence to find the next number.
• Here’s an example of a sequence:
  • If we say $a_n = a_{n-1} + 2$ and start with $a_1 = 3$, we can find:
    • The first number is 3.
    • The second number is 5 (3 + 2).
    • The third number is 7 (5 + 2).
    • The fourth number is 9 (7 + 2).

So, the sequence goes: 3, 5, 7, 9, ...

Explicit Formulas

• This type gives us a direct way to find any number in the sequence without needing the previous ones.
• For the same sequence we talked about, we can use this formula: $a_n = 2n + 1$.
• This means:
  • For the first number ($n=1$), we get $2(1) + 1 = 3$.
  • For the second number ($n=2$), it’s $2(2) + 1 = 5$.
  • For the third number ($n=3$), it’s $2(3) + 1 = 7$.
  • And for the fourth number ($n=4$), we find $2(4) + 1 = 9$.

Comparison

• Recursive formulas: You need to know previous numbers to find the next one.
• Explicit formulas: You can calculate any number directly.

Both methods will give you the same results, but they are just different ways to find the numbers in a sequence!