Question 53
Several natural numbers are given on a line. We perform a transformation as follows: for every pair of consecutive integers on the line, write the sum of those two numbers in the middle of them. After 2013 such steps, how many times number 2013 are there on the line if
a) The given numbers are $1$ and $1000$?
b) The given numbers are $1, 2, 3, \ldots, 1000$ on the increasing order from the left to the right?
a) The given numbers are $1$ and $1000$?
b) The given numbers are $1, 2, 3, \ldots, 1000$ on the increasing order from the left to the right?