Input a number to check if it's a Lucas number, or specify an Nth term to calculate the value and sum of the sequence up to that point.
Lucas Sequence Calculate
Result
N-th Term
Sum
What is a Lucas Sequence?
The Lucas Sequence is a type of recursive number series defined as:
\( L_0 = 2 \) (0th term)
\( L_1 = 1 \) (1st term)
For \( n \geq 2 \), \( L_n = L_{n-1} + L_{n-2} \) (each term is the sum of the two preceding terms).
Numbers in this sequence are called Lucas numbers.
How to Check if a Number is a Lucas Number
Generate the Sequence: Start with \( L_0 = 2 \) and \( L_1 = 1 \), and recursively calculate subsequent terms until reaching or exceeding the number in question.
Compare: If the number matches one of the generated terms, it is a Lucas number. Otherwise, it is not.
Calculating the Nth Lucas Number and Total Sum
Find the Nth Term: Use the recursive formula \( L_n = L_{n-1} + L_{n-2} \) to calculate the Nth term.
Calculate the Total Sum: Add all terms from \( L_0 \) to \( L_n \) to find the cumulative sum.
Examples
Example 1: Is 11 a Lucas Number?
Solution:
Generate the sequence:
\(L_0 = 2\)
\(L_1 = 1\)
\(L_2 = 3\)
\(L_3 = 4\)
\(L_4 = 7\)
\(L_5 = 11\)
Result: Since 11 is in the sequence, it is a Lucas number.
Example 2: Is 15 a Lucas Number?
Solution:
Generate the sequence:
\(L_0 = 2\)
\(L_1 = 1\)
\(L_2 = 3\)
\(L_3 = 4\)
\(L_4 = 7\)
\(L_5 = 11\)
\(L_6 = 18\)
Result: 15 is not in the sequence, so it is not a Lucas number.
Example 3: Calculate the 5th Term and Total Sum of the Lucas Sequence