Certainly! Let's explore how initial conditions and sequence requirements can change whether you use $f(n-1)$ or $f(n-2)$, and provide examples with tables.

Example 1: Initial Conditions and Recursive Relations

Case 1: Using $f(n-1)$

Consider the simple arithmetic sequence defined by: $f(n) = f(n-1) + 2$ Initial Condition: $f(1) = 3$

This sequence is an arithmetic progression where each term increases by 2 from the previous term.

• Recursive Relation: $f(n) = f(n-1) + 2$
• Initial Condition: $f(1) = 3$

Sequence: 3, 5, 7, 9, 11, ...

• $f(2) = f(1) + 2 = 3 + 2 = 5$
• $f(3) = f(2) + 2 = 5 + 2 = 7$

Why only $f(n-1)$?

• Each term only depends on the immediately previous term.

Case 2: Using $f(n-1)$ and $f(n-2)$

Now, consider the Fibonacci sequence: $f(n) = f(n-1) + f(n-2)$ Initial Conditions: $f(1) = 1$, $f(2) = 1$

This sequence is defined by each term being the sum of the two preceding terms.

• Recursive Relation: $f(n) = f(n-1) + f(n-2)$
• Initial Conditions: $f(1) = 1$, $f(2) = 1$

Sequence: 1, 1, 2, 3, 5, 8, 13, ...

• $f(3) = f(2) + f(1) = 1 + 1 = 2$
• $f(4) = f(3) + f(2) = 2 + 1 = 3$

Why $f(n-2)$?

• Each term depends on the sum of the two previous terms.

Example 2: Tables Requiring Different Equations

Table 1: Using $f(n-1)$

$n$	$f(n)$
1	4
2	6
3	8
4	10
5	12

This table represents a simple arithmetic sequence where each term increases by 2: $f(n) = f(n-1) + 2$ Initial Condition: $f(1) = 4$

• $f(2) = 4 + 2 = 6$
• $f(3) = 6 + 2 = 8$

Why $f(n-1)$?

• The pattern is consistent with each term only depending on the previous term.

Table 2: Using $f(n-1)$ and $f(n-2)$

$n$	$f(n)$
1	1
2	1
3	2
4	3
5	5

This table represents the Fibonacci sequence: $f(n) = f(n-1) + f(n-2)$ Initial Conditions: $f(1) = 1$, $f(2) = 1$

• $f(3) = 1 + 1 = 2$
• $f(4) = 2 + 1 = 3$

Why $f(n-2)$?

• The pattern shows that each term depends on the sum of the two preceding terms.

Summary:

• Single Dependency: Use $f(n-1)$ when each term only depends on the previous term.
• Multiple Dependencies: Use $f(n-1)$ and $f(n-2)$ (or more) when the sequence depends on multiple preceding terms.

Would you like to explore more examples or further explanations?

Here are some related questions:

1. Can you show an example where $f(n)$ depends on $f(n-3)$?
2. What are the implications of higher-order recursions in computing?
3. How do initial conditions affect the uniqueness of a sequence?
4. Can you give a real-world application of the Fibonacci sequence?
5. How do non-linear recurrence relations differ from linear ones?
6. What happens if initial conditions are altered in a recursive sequence?
7. Can a sequence involve both $f(n-1)$ and $f(n+1)$?
8. How can we solve a recurrence relation using generating functions?

Tip: When working with recursive sequences, always write down the first few terms explicitly to understand the pattern before generalizing the recursive formula.