The sequence 1,1,2,3,5,8,13,21 has the property that each term (starting with the third term) is the sum of the previous two terms. How many of the first 1000 terms are divisible by 4?
This sequence is called the Fibonacci sequence.
Let's write down and study the sequence of remainders
of the Fibonacci numbers modulo 4.
We have $1, 1, 2, 3, 1, 0, 1, 1, \dots$ so this sequence
of remainders is periodic where every 6th term only is divisible by 4.
Therefore, of the first 1000 Fibonacci numbers, $\lfloor 1000/6 \rfloor = 166$ of them are divisible by 4.