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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)

- 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 \(k \geq 2\)

Logic

Sequence

Composite number

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

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.

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.

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If and only if k>=2 and k-2 is divisible by 3