Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Sequence and composite number.
Consider the sequence \(a_1\)=101, \(a_2\)=10101,\(a_3\)=1010101 and so on. Then \(a_k\) is a composite number ( that is not a prime number)
Logic
Sequence
Composite number
Answer: if and only if \(k \geq 2\) and k-2 is divisible by 3
B.Stat Objective Question 75
Challenges and Thrills of Pre-College Mathematics by University Press
for \(a_k\) \(k \geq 2\) may be prime also then not considering this here
for \(a_{8}\) \(10^{9}-1\) and \(10^{9}+1\) not divisible by 11
8-2 is divisible by 3 and \(a_{8}\) is composite number then \(a_{k}\) is composite if and only if \(k \geq 2\) and k-2 is divisible by 3.
I.S.I. & C.M.I. Entrance Program
Taught by students and alumni of I.S.I. & C.M.I.
If and only if k>=2 and k-2 is divisible by 3