Question 549912: select a natural number between 1 to 50. square the digits and add. repeat this process until you see a pattern.
What other numbers end with one?
What happen if you do the process to number 37?
What conclusion can you give?
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Suppose I choose 4. The numbers in the pattern are 16, 37, 58, 89, 145, 42, 20, 4, 16, and it repeats.
If the number 1 appears in the sequence, then the previous number must have been 1, 10, 100 or any power of 10. However, all succeeding terms in the sequence must be 1.
Doing the process to the number 37 yields the same sequence as if I did 4 (up to shifting terms).
We can conjecture that every number will eventually produce a periodic sequence. However we want to prove this. Given a sequence a_1, a_2, ... we want to show that a_i = a_j for some distinct i,j (this will produce a periodic sequence). Since it is tricky to establish an explicit formula for a_i, we can instead show that a_i attains a maximum value. Since your sequence can be arbitrarily long, then by Pigeonhole principle, two numbers in the sequence must be the same, and hence the sequence is periodic. I'll leave it to you to prove that a_i attains a maximum value...
|
|
|
