Recursive notation for exponential models
Recursive Notation for Exponential Models
Recursive Notation for Exponential Models is a topic in Exponential & Logarithmic Functions in the Common Core State Standards. It is aligned to Standard F-BF.2, which requires students to write arithmetic and geometric sequences both recursively and with an explicit formula, including exponential models.
A recursive formula defines each term from the previous one, \(a_n=r\,a_{n-1}\); a constant ratio makes it exponential with \(a_n=a_0 r^n\).
Theory
A recursive formula builds a sequence from the term before it plus a starting value:
A constant ratio \(r\) makes it exponential, with explicit form \(a_n=a_0\cdot r^{n}\).
Recursive and explicit:
How to use recursion
- Identify the starting value \(a_0\).
- Find the constant ratio \(r\).
- Write \(a_n=r\,a_{n-1}\) (recursive).
- Convert to \(a_n=a_0 r^n\) for direct evaluation.
Multiply by \(2\) each step.
| \(a_0,a_1,a_2,a_3\) | \(=\) | \(3,\ 6,\ 12,\ 24\) |
A constant ratio \(2\) gives an exponential.
| \(a_n\) | \(=\) | \(3\cdot2^{n}\) |
Each day is twice the last.
| \(P_0\) | \(=\) | \(50\) |
| \(P_n\) | \(=\) | \(2P_{n-1}\) |
Ratio \(1.5\), initial \(100\).
| \(a_n\) | \(=\) | \(100(1.5)^{n}\) |
Common pitfalls
Frequently asked questions
What is a recursive formula?
One that defines each term from the previous term and a starting value.
How is it different from an explicit formula?
Explicit gives any term directly; recursive builds up term by term.
When is a recursive sequence exponential?
When each term is a constant multiple of the previous one.
How do you convert \(a_n=r\,a_{n-1}\) to explicit form?
\(a_n=a_0\,r^n\).