Recursive Formula - Algebra-1

1. Fundamental Concepts

Definition: A recursive formula is a way to describe a sequence where each term is defined using one or more preceding terms (usually the previous term). For arithmetic sequences, the core of the recursive formula is to establish the relationship between terms using the common difference (denoted as d, the difference between consecutive terms).
Components:
- Initial Term: The first term of the sequence, usually denoted as \(a_1\), which is the starting point of the recursive formula.
- Recursive Relation: Specifies the relationship between the n-th term (\(a_n\)) and the previous term (\(a_{n-1}\)). For arithmetic sequences, this relationship is \(a_n = a_{n-1} + d\) (where \(n \geq 2\)).

Difference from Explicit Formula: The explicit formula allows direct calculation of the n-th term using the term number n, with the form \(a_n = a_1 + (n - 1)d\). In contrast, the recursive formula requires knowing the previous term to calculate the current term, relying more on the "recursive nature" of the sequence.
Applicable Scenarios: Recursive formulas are suitable when the initial term and common difference are known, and there is a need to derive subsequent terms step by step. They are particularly convenient when the number of terms is small or when observing the generation process of the sequence.
Role of the Common Difference: The common difference d is crucial in the recursive formula of an arithmetic sequence. It determines the difference between each term and the previous one, serving as the core constant that maintains the recursive relationship.

Easy Level: Given an arithmetic sequence with the first term \(a_1 = 5\) and common difference \(d = 3\), write its recursive formula and find the 3rd term.
- Recursive Formula: \(a_1 = 5\); \(a_n = a_{n-1} + 3\) (\(n \geq 2\))
- Solution: \(a_2 = a_1 + 3 = 5 + 3 = 8\); \(a_3 = a_2 + 3 = 8 + 3 = 11\)
Medium Level: The 2nd term of an arithmetic sequence is 10, and the common difference \(d = 4\). Write the recursive formula and find the 5th term.
- First, find the first term: \(a_1 = a_2 - d = 10 - 4 = 6\)
- Recursive Formula: \(a_1 = 6\); \(a_n = a_{n-1} + 4\) (\(n \geq 2\))
- Solution: \(a_3 = a_2 + 4 = 10 + 4 = 14\); \(a_4 = a_3 + 4 = 14 + 4 = 18\); \(a_5 = a_4 + 4 = 18 + 4 = 22\)
Hard Level: Given that in an arithmetic sequence, \(a_3 = 15\) and \(a_5 = 23\), write the recursive formula and find the 8th term.
- First, find the common difference: Since \(a_5 = a_3 + 2d\), we have \(23 = 15 + 2d\), so \(d = 4\)
- Find the first term: \(a_3 = a_1 + 2d\), that is, \(15 = a_1 + 2 \times 4\), so \(a_1 = 7\)
- Recursive Formula: \(a_1 = 7\); \(a_n = a_{n-1} + 4\) (\(n \geq 2\))
- Solution: \(a_6 = a_5 + 4 = 23 + 4 = 27\); \(a_7 = a_6 + 4 = 27 + 4 = 31\); \(a_8 = a_7 + 4 = 31 + 4 = 35\)

Identify the Initial Term and Common Difference: When solving problems related to recursive formulas, first determine the first term \(a_1\) and the common difference d of the sequence. If they are not given directly, they can be derived from known terms (e.g., find the common difference using the difference between two terms, then find the first term based on the relationship between term numbers).
Derive Step by Step: Use the recursive formula \(a_n = a_{n-1} + d\) to calculate subsequent terms one by one starting from the initial term, which is especially suitable for cases with a small number of terms.
Verify Rationality: After calculating the results, verify them using the explicit formula to ensure that each step of the recursive derivation conforms to the properties of arithmetic sequences (the difference between consecutive terms is the common difference).
Distinguish Between Recursive and Explicit Scenarios: The explicit formula is more efficient when there is a need to quickly find a distant term (e.g., the 100th term). However, the recursive formula is more suitable when there is a need to generate the sequence step by step or analyze the dependency between terms.