Question

Here is a function that describes a sequence called the Lucas numbers, which is similar to the Fibonacci sequence:

f(n) = f(n-1) + f(n - 2); f(1) = 2; f(2) = 1.
• Describe in words the relationship between the terms created by this algebraic rule.
• List the first six terms of the sequence.
• Find the 12th term.

Answer 1

The Lucas numbers are a sequence where each term is the sum of the two preceding terms. The first six terms are 2, 1, 3, 4, 7, 11. The 12th term is 199.

Step-by-step explanation:

The Lucas numbers are a sequence similar to the Fibonacci sequence, where each term is the sum of the two preceding terms. The relationship is described by the recursive formula f(n) = f(n-1) + f(n-2).

The first six terms of the sequence are: 2, 1, 3, 4, 7, 11.

To find the 12th term, we can use the recursive formula f(n) = f(n-1) + f(n-2) repeatedly until we reach the 12th term. Starting from f(1) = 2 and f(2) = 1, we can calculate f(12) as follows:

1. f(3) = f(2) + f(1) = 1 + 2 = 3
2. f(4) = f(3) + f(2) = 3 + 1 = 4
3. f(5) = f(4) + f(3) = 4 + 3 = 7
4. f(6) = f(5) + f(4) = 7 + 4 = 11
5. f(7) = f(6) + f(5) = 11 + 7 = 18
6. f(8) = f(7) + f(6) = 18 + 11 = 29
7. f(9) = f(8) + f(7) = 29 + 18 = 47
8. f(10) = f(9) + f(8) = 47 + 29 = 76
9. f(11) = f(10) + f(9) = 76 + 47 = 123
10. f(12) = f(11) + f(10) = 123 + 76 = 199

Therefore, the 12th term of the Lucas numbers sequence is 199.