1. If we take this sequence mod 2, we get 1,1,0,1,1,0,1,1,0,1,1,0, so it has period 3. To prove this, we can first see that taking mod 2 is basically seeing if the numbers are odd or even. Since the first two are odd, the third must be even because the sum of 2 odds is always even. The term after that must be odd, since is it a sum of an even number and an odd number. The term after that must be odd as well, since it is a sum of an even number and an odd number. The last two terms are both odd, just like the two ones at the start of the sequence, so we know that this will repeat.
Next we can take this sequence mod 3, to get 1,1,2,0,2,2,1,0,1,1,2,0,2,2,1. This sequence has period 8. This is the case because we can just add up each term of the mod sequence and take mod 3. Every time 1,1 comes up, the sequence must repeat.
We can do the same with mod 5, and just count the number of terms until 1,1 comes up again.. 1,1,2,3,0,3,3,1,4,0,4,4,3,2,0,2,2,4,1,0, 1,1,2,3,0, which has period 20.