List the first eight terms of the sequence formed by finding the differences of successive terms in the Fibonacci sequence

Question

List the first 8 terms of the sequence formed by finding the differences of successive terms

Answer 1

Hello Brandi,

Since the Fibonacci sequence is recursively defined by

a₀ = 1, a₁ = 1;  a_n+2 = a_n+1 + a_n for n ≥ 0,

we find that the sequence you are interested in, let's call it {b_n}, satisfies

b₁ = a₁ - a₀ = 0;  b_n+2 = a_n+2 - a_n+1 = a_n, for n ≥ 0,

by the definition of the Fibonacci sequence.  Therefore, {b_n} is another Fibonacci-type sequence itself, formed by the seeds (0,1), rather (1,1) as in the "usual" case.  

Another way of putting this is that your sequence is a shift of the Fibonacci sequence, with 0 to occupy the first (newly created) slot.

Regards,

Hassan H.