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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

But try the problem first…

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

Source

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.