Question

4/2/25. An infinite sequence of real numbers a1, a2, a3, . . . is called spooky if a1 = 1 and for
all integers n > 1,
na1 + (n−1)a2 + (n−2)a3 +···+ 2an−1 +an <0, n2a1 +(n−1)2a2 +(n−2)2a3 +···+22an−1 +an >0.
Given any spooky sequence a1, a2, a3, . . ., prove that
20133a1 + 20123a2 + 20113a3 + · · · + 23a2012 + a2013 < 12345