Try this TOMATO problem from I.S.I. B.Stat Entrance Objective Problem based on Sequence and composite number.

Composite number Problem (B.Stat Objective)


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)

  • if and only if \(k \geq 2\) and \(11|(10^{k+1}+1)\)
  • if and only if \(k \geq 2\) and k-2 is divisible by 3
  • if and only if \(k \geq 2\) and \(11|(10^{k+1}-1)\)
  • if and only if \(k \geq 2\)

Key Concepts


Logic

Sequence

Composite number

Check the Answer


But try the problem first…

Answer: if and only if \(k \geq 2\) and k-2 is divisible by 3

Source
Suggested Reading

B.Stat Objective Question 75

Challenges and Thrills of Pre-College Mathematics by University Press

Try with Hints


First hint

for \(a_k\) \(k \geq 2\) may be prime also then not considering this here

Second Hint

for \(a_{8}\) \(10^{9}-1\) and \(10^{9}+1\) not divisible by 11

Final Step

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.

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