Question 1053445
Base case n = 1:
3 divides *[tex \large 4^1 - 1], which is correct.


Inductive step: n = k+1:
Suppose 3 divides *[tex \large 4^k - 1] for some *[tex \large k \ge 1]. Then *[tex \large 4^k - 1 = 3m] for some integer m. Multiply by 4 and we have that *[tex \large 4^{k+1} - 4 = 12m]. Add 3 to both sides to get *[tex \large 4^{k+1} - 1 = 12m + 3 = 3(4m+1)]. Since the right-hand side is a multiple of 3, so is the left-hand side. Therefore 3 divides *[tex \large 4^{k+1} - 1].