Fibonacci Number
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
Start with this mental model
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
How it works, without unnecessary jargon
Naive recursion recomputes the same subproblems and grows exponentially.
An iterative solution keeps only the previous two numbers, giving O(n) time and O(1) extra space.
Read the algorithm as a sequence of decisions
State the mathematical identity/property.
Choose numeric representation and constraints.
Translate the property into an iterative/recursive computation and analyze precision.
Read a tiny input by hand and write down the state after every important operation. For Fibonacci Number, the goal is to make the invariant visible rather than to memorize code.
function fibonacci(n) {
let a = 0, b = 1;
for (let i = 0; i < n; i++) {
[a, b] = [b, a + b];
}
return a;
}Do not memorize the syntax. Follow the state that changes on each line and connect it to the invariant described above.
O(n) iterative / O(2ⁿ) naive recursion
Do not quote this complexity mechanically. Ask which operation or input dimension it describes, then derive the cost from the loops, recursion depth, data-structure operations and any preprocessing assumptions.
Good fit
- Use the direct mathematical property when it removes unnecessary brute force.
What usually goes wrong
- Ignoring floating-point precision or integer range.
- Using a formula outside its valid assumptions.
A professional answer shape
Use Fibonacci to explain overlapping subproblems, memoization and the difference between elegant-looking recursion and efficient recursion.
What problem does Fibonacci Number solve?
The Fibonacci sequence starts from base values and each next number is the sum of the previous two.
What should I mention about complexity?
Its typical cost here is O(n) iterative / O(2ⁿ) naive recursion, but a complete answer should connect that cost to the representation, input constraints and the dominant operations.
What is the most common mistake?
Ignoring floating-point precision or integer range.