Can anyone help me in this question:
Define $ (b_n)$ as $b_1= 1,b_n=a_{n+1} - a_n $ for $ n\ge 2$, where $ a_n $ is the Fibonnaci series. This sequence is known as the sequence of Lucas numbers.
Prove:
(i) $b_n=b_{n-1} + b_{n-2}$ for $ n \ge 3$
Here is what I have done:
$(a_n)=1,1,2,3,5,....$
$b_2=a_3 -a_2=2-1=1 $
I have to prove the equation through mathematical induction. So:
LHS
$(b_3)=a_4 -a_3=3-2=1 $
RHS
$b_2 + b_1 = 1 +1 =2 $
But here the LHS and RHS should be equal. So I looked up on wikipedia and there it says Lucas numbers are $2,1,3,4,7,11,18,... $
Can anyone clarify what is happening here?
From definition $$b_{n-1}+b_{n-2}=a_n-a_{n-1}+a_{n-1}-a_{n-2}=a_{n-1}=a_{n+1}-a_n=b_n$$The last two steps follow from the definition of Fibonacci sequence and definition of $b_n$. Thus $b_n$ is also a Fibonacci number.