Key takeaway: A recursive formula builds a sequence one term at a time, using the term or terms just before to produce the next one, and it must state where the sequence starts.
Grade 9 – Algebra 1F-BF.A.2Functions and Graphs
Definition of Recursive Formula
A recursive formula works like instructions for a chain reaction: “start here, then do this to the last number to get the next one.” It has two parts. The first part gives the starting term, such as \(a_1 = 4\). The second part is the rule that links a term to the previous term.
To find a distant term, you must compute every term before it. That makes a recursive rule excellent for describing how a pattern grows step by step, such as a savings balance updated each month, but slow for jumping straight to the 100th term. An explicit formula can do that jump.
Formula and notation
The previous term is written \(a_{n-1}\). A typical rule is
\[a_1 = 4, \qquad a_n = 3a_{n-1} – 5 \ \ (n \ge 2)\]
Never omit the starting value: without it, the rule cannot begin. The condition \(n \ge 2\) says where the rule applies.
Key facts
- Every recursive formula needs a first term and a rule for finding the next one.
- A rule that adds the same number each step produces an arithmetic pattern.
- A rule that multiplies by the same number each step produces a geometric pattern.
- Reaching term 50 recursively means computing terms 1 through 49 first.
Recursive Formula example
A pond-restoration group plants by the rule \(a_1 = 4\) and \(a_n = 3a_{n-1} – 5\), where \(a_n\) is the number of dozens of reeds in week \(n\). Find the first five terms.
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Start: \(a_1 = 4\).
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\(a_2 = 3(4) – 5 = 7\) and \(a_3 = 3(7) – 5 = 16\).
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\(a_4 = 3(16) – 5 = 43\) and \(a_5 = 3(43) – 5 = 124\).
The first five terms are 4, 7, 16, 43, 124.
Common mistake
Watch out
Hana computes \(a_3\) as \(3a_1 – 5 = 3(4) – 5 = 7\), reusing the first term at every step.
Why it is wrong
The rule says to use the term just before. For \(a_3\) that is \(a_2 = 7\), not \(a_1\): \(a_3 = 3(7) – 5 = 16\). Each new term becomes the input for the next step.
Related terms and lessons
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