Choose any positive integer. powers of two here are not very interesting, so choose something else. if the number you have chosen is even, divide it by two. if it's odd, multiply it by three and add one. now, with the result, repeat the same process. if the number is even divide it by two. if it's odd multiply it by three and add one. keep repeating this process on the outcome. you will generate (*recursively*) a sequence of positive integers.

Step-by-step explanation:

Let the integer be 6 for even and 7 for odd (say)

For 6, we divide by 2, now get 3.  Now we multiply by 3 and add 1 to get 10. Now since 10 is even divide by 5, now multiply by 3 and add 1 to get 16.  Now divide by 2 again by 2 again by 2 again by 2 till we get rid of even numbers.

The result is 1, so multiply by 3 and add 1 we get 4 now divide 2 times by 2 to get 1, thus this result now again repeats after 2 times.

Say if we select off number 3, multiply by 3 and add 1 to get 10 now divide by 5, now repeat the same process as above for 5 until we get 1 and it gets repeated every third time.

Thus whether odd or even after some processes, we get 1 and the process again and again returns to 1.

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