# determine the number of terms in a fibonacci sequence that are divisible by $3$

Consider the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, 21, . . .$ where each term, after the first two, is the sum of the two previous terms. How many of the first $1000$ terms are divisible by 3?

• The fact $(F_a,F_b)=F_{(a,b)}$ (so $F_a\mid F_b$ if $a\mid b$) might be helpful. – Hanul Jeon Sep 24 '14 at 15:16
• Try dividing the first few by $3$ to see what happens. If you spot a pattern, then prove it persists. – Mark Bennet Sep 24 '14 at 15:18

HINT : In mod $3$, $$\color{red}{1},1,2,0,2,2,1,0,\color{red}{1},1,2,0,\cdots$$