Recursive Formula
In short: A recursive formula defines each term of a sequence using the term just before it, together with a starting value that gets the list going.
Algebra 1Functions and GraphsF-BF.A.2
What is Recursive Formula?
Imagine climbing stairs where each step tells you how far to go from the one you stand on; a recursive formula works that way, giving the first term and then a rule that connects every term to the previous one.
For example, \(a_1 = 3\) and \(a_n = 2a_{n-1} – 1\) means start at 3, then double the last term and subtract 1: 3, 5, 9, 17, 33. You must compute the terms in order, because each one depends on its predecessor.
How to spot it
- A rule says next term equals the previous term plus, minus or times something.
- A starting value such as \(a_1 = 7\) is given along with a pattern.
- The subscripts \(n\) and \(n – 1\) appear in the same equation.
- A situation describes what happens each step based on what you had before, like a balance after each deposit.
The method
- Write the starting value \(a_1\), which the formula must always include.
- Substitute \(n = 2\) into the rule and use \(a_1\) to compute \(a_2\).
- Repeat with \(n = 3\), \(n = 4\) and so on, each time using the term you just found.
- To build a formula from a list, check whether you add or multiply to get from one term to the next.
Recursive Formula: solved example
Problem
A toy tower uses 4 blocks at level 1. A recursive rule says \(a_1 = 4\) and \(a_n = 3a_{n-1} – 2\). Find the number of blocks at level 4.
Solution
\(a_2 = 3(4) – 2 = 10\). Then \(a_3 = 3(10) – 2 = 28\).
Finally \(a_4 = 3(28) – 2 = 82\). Level 4 uses 82 blocks.
Try it
Question 1
List the first four terms if \(a_1 = 7\) and \(a_n = a_{n-1} – 4\).
Show the answer
\(7, 3, -1, -5\).
Question 2
Write a recursive formula for \(2, 6, 18, 54, \ldots\).
Show the answer
Each term is 3 times the previous one: \(a_1 = 2\) and \(a_n = 3a_{n-1}\).
Recursive Formula vs. Explicit Formula
A recursive rule needs the previous term, so to reach the 50th you must grind through 49 steps. The explicit kind jumps straight to any term from its position n alone. Recursive rules describe how a pattern grows; explicit ones are faster for answers far down the list.
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