Let us have a series of numbers 16,1156,111556, .... . The series is formed by k ones, (k-1) fives and one 6 for k>0. Then each number of the series is a
A) Prime
B) Perfect Square
C) Prime Square
D) Atleast one of the foregoing
E) None of the foregoing
Tk = (11....k)*10^k + (55...k-1)*10 +6
ReplyDelete= [(10^k-1)*10^k + 5(10^k-1 - 1)*10 + 54]/9
= [10^2k +4*10^k +4]/9
=(10^k+2)^2/9
=[(10^k+2)/3]^2
(10^k+2) will always be divisible by 3. Hence each term of the series is perfect square.
Apart from that tk = [(3....k-1times)*10+4]^2. Hence only B could be the answer