Enter the Nth term to calculate the corresponding Pell number. The Pell number sequence is defined by the recurrence relation Pn = 2Pn-1 + Pn-2, with P0 = 0 and P1 = 1.
Pell Number Formula
The Pell number for the Nth term is calculated using the following formula:
P<sub>n</sub> = 2P<sub>n-1</sub> + P<sub>n-2</sub>
Variables:
- Pn is the Pell number at the Nth term
- Pn-1 is the Pell number at the (N-1)th term
- Pn-2 is the Pell number at the (N-2)th term
To calculate the Pell number for the Nth term, use the recurrence relation to iteratively compute the value starting from the base cases P0 = 0 and P1 = 1.
What is a Pell Number?
Pell numbers are a sequence of integers that are solutions to the Pell equation, a specific type of Diophantine equation. The sequence begins with P0 = 0 and P1 = 1, and each subsequent number is the sum of twice the previous Pell number and the Pell number before that. Pell numbers have applications in number theory and the approximation of square roots.
How to Calculate Pell Numbers?
The following steps outline how to calculate Pell numbers:
- Start with the base cases: P0 = 0 and P1 = 1.
- For N > 1, use the recurrence relation Pn = 2Pn-1 + Pn-2 to calculate the Nth Pell number.
- Iterate this process until the Nth term is reached.
- Use the calculator above to verify your results.
Example Problem:
Calculate the 10th Pell number.
Using the recurrence relation:
P10 = 2P9 + P8
With P9 and P8 previously calculated as part of the sequence, P10 can be found. Insert the values into the calculator above to find the 10th Pell number.
