RECURSIVE FORMULA

Recursive Formula

Definition Of Recursive Formula

Recursive formula is a formula that is used to determine the next term of a sequence using one or more of the preceding terms.

Examples of Recursive Formula

The recursive formula for the sequence 5, 20, 80, 320, ... is a_n = 4a_n-1.

Video Examples: Recursive Formula

Solved Example on Recursive Formula

Ques: The first term in a sequence is 39. Each term after the first term is 7 more than the term before it. Find the recursive formula for the sequence.

Choices:

A. a_n= a_(n-1) / 7 where a₁ = 39
B. aa_n= a_(n-1) - 7, where a₁ = 39
C. aa_n=7a_(n-1) - 7, where a₁ = 39
D. aa_n= a_(n-1) + 7, where a₁ = 39
Correct Answer: D

Solution:

Step 1: a 1 = 39 [First term = 39.]
Step 2: a 2 = a 1 + 7 = 39 +7 = 46 [Each term is 7 more than the term before it.]
Step 3: a 3 = a 2 + 7 = 46 + 7 = 53
Step 4: an = a_(n-1) + 7 [This is a recursive formula.]
Step 5: The recursive formula for the sequence is an = a_(n-1) + 7, where a 1 = 39.

Quick Summary

Defines a term based on previous terms.
Requires initial term(s) to start the sequence.
Useful for defining patterns where each term depends on its predecessor.

\[ a_n = f(a_{n-1}) \]

🍎 Teacher Insights

Emphasize the importance of the initial term(s) and provide plenty of examples to illustrate the concept.

🎓 Prerequisites

Basic Algebra
Understanding of Sequences

Check Your Knowledge

Q1: What is the next term in the sequence defined by a_n = 2a_{n-1}, where a_1 = 3?

5 6 9 12

Q2: Which of the following is a recursive formula for the sequence 1, 3, 5, 7, ...?

a_n = 2n - 1 a_n = a_{n-1} + 2, a_1 = 1 a_n = n + 2 a_n = a_{n-1} * 2, a_1 = 1

Frequently Asked Questions

Q: What is the difference between a recursive and explicit formula?
A: A recursive formula defines a term based on previous terms, while an explicit formula defines a term directly in terms of its position in the sequence (n).

Q: How do I find a recursive formula?
A: Look for a pattern relating each term to the one(s) before it. Express this relationship as a formula.