21) A kangaroo climbs a staircase of 5000 steps in this way: first it jumps to the $3^{\text{rd}}$ step, then back 1, then up 5, back 3, forward 7, down 5, and so on. Unfortunately, one step is wobbly, and if the kangaroo jumps on it, it will fall to the ground. Will the kangaroo manage to climb to the top if the wobbly step is:
(A) the $2000^{\text{th}}$
(B) the $2001^{\text{st}}$
(C) the $2002^{\text{nd}}$
(D) the $2003^{\text{rd}}$
21) The answer is (B). The sequence of jumps of the kangaroo is $3-1+5-3+7-5+\ldots=(3-1)+(5-3)+(7-5)+\ldots=3+(-1+5)+(-3+7)+(-5+9)+\ldots$ Therefore, it will jump on all even steps (precisely after the jump $2 k$ it will be on step $2 k$), while it will touch all and only the odd steps of the form $4 k+3$. (precisely after the jump $1+2 k$ it will be on step $3+4 k$). Thus, it will be able to climb to the top only if the dangerous step is of the form $4 k+1$.