Algebra 2
Exponential and logarithmic functions
Recursive notation for exponential models
20 practice questions
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Theory + worked examples
Theory
A recursive formula builds a sequence from the term before it plus a starting value:
\[a_n=r\cdot a_{n-1},\qquad a_0\text{ given}.\]
A constant ratio \(r\) makes it exponential, with explicit form \(a_n=a_0\cdot r^{n}\).
Recursive gives the next term; explicit gives any term directly.
Each term is twice the one before.
Recursive vs explicit form.
Recursive and explicit:
\[a_n=r\,a_{n-1},\qquad a_n=a_0\,r^{n}\]
The ratio \(r\) is the growth or decay factor.
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.
Example 1 — List the terms
For \(a_0=3,\ a_n=2a_{n-1}\), list the first four terms.
Solution
Multiply by \(2\) each step.
| \(a_0,a_1,a_2,a_3\) | \(=\) | \(3,\ 6,\ 12,\ 24\) |
Example 2 — Explicit formula
Write an explicit formula for that sequence.
Solution
A constant ratio \(2\) gives an exponential.
| \(a_n\) | \(=\) | \(3\cdot2^{n}\) |
Example 3 — Model a population
A colony doubles daily from \(50\). Write a recursive model.
Solution
Each day is twice the last.
| \(P_0\) | \(=\) | \(50\) |
| \(P_n\) | \(=\) | \(2P_{n-1}\) |
Example 4 — Recursive to explicit
Convert \(a_0=100,\ a_n=1.5a_{n-1}\) to explicit form.
Solution
Ratio \(1.5\), initial \(100\).
| \(a_n\) | \(=\) | \(100(1.5)^{n}\) |
Common pitfalls
State the starting value — recursion needs \(a_0\) or \(a_1\).
Recursive needs the previous term; explicit does not.
A constant ratio (not difference) makes it exponential.
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?
\(a_n=a_0\,r^n\).
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