Calculate the sum of the first N Lucas numbers instantly. Learn the Lucas sequence, the formula for its sum, and the relationship between Lucas and Fibonacci numbers.
Enter \(N\) to calculate:
Enter a number from 1 to 10,000. For large N, the calculator calculates the exact result without converting the Lucas number to a normal PHP integer.
Lucas numbers are a sequence of integers closely related to the Fibonacci sequence.
The Lucas sequence starts with:
Each subsequent Lucas number is calculated by adding the previous two:
The beginning of the Lucas sequence is:
If the Lucas sequence starts with \(L_0=2\) and \(L_1=1\), the sum of the first \(N\) terms is:
For example, the first five Lucas numbers are:
Their sum is:
Using the identity:
The recurrence relation can be rearranged as:
Therefore:
Most of the terms cancel, leaving only \(L_{N+1}-1\).
Lucas and Fibonacci numbers use the same recurrence:
Their starting values are different.
| Sequence | First term | Second term | Beginning |
|---|---|---|---|
| Lucas | 2 | 1 | 2, 1, 3, 4, 7, 11... |
| Fibonacci | 0 | 1 | 0, 1, 1, 2, 3, 5... |
This difference in starting values produces two related but distinct sequences.
Lucas numbers are a sequence beginning with 2 and 1. Each subsequent term is the sum of the previous two terms.
With \(L_0=2\) and \(L_1=1\), the sum of the first \(N\) Lucas numbers is:
The first five Lucas numbers are \(2,1,3,4,7\). Their sum is:
With the standard indexing used here, the first Lucas number is \(L_0=2\), followed by \(L_1=1\).
Both sequences use the same recurrence relation, but their starting values differ. Lucas numbers start with 2 and 1, while Fibonacci numbers conventionally start with 0 and 1.
For the first \(N\) Lucas numbers, \(L_0\) through \(L_{N-1}\), the sum is: