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Arithmetic presentation

Arithmetic presentation. By: Alexandra Silva & Dani Hoover. Goals :. Does T n converge to a fixed point for all n? Does it always take the same number of steps? Can we make any generalizations for T n ?. T 2. Example: Start with the number 49 94-49=45 54-45=09 90-09=81 81-18=63

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Arithmetic presentation

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  1. Arithmetic presentation By: Alexandra Silva & Dani Hoover

  2. Goals: • Does Tn converge to a fixed point for all n? • Does it always take the same number of steps? • Can we make any generalizations for Tn?

  3. T2 • Example: Start with the number 49 94-49=45 54-45=09 90-09=81 81-18=63 63-36=27 72-27=45 54-45=09 Example 2: Start with the number 24 42-24=18 81-18=63 63-36=27 72-27=45 Pattern starts to repeat!

  4. T3 • Example: Start with the number 123 321-123=197 971-179=792 972-279=693 963-369=594 954-459=495 954-459=495 *Therefore, we conclude that all T3 (with the exception of aaa) converge to a fixed point 495.

  5. T4 • Example: Start with the number 2143 4321-1234=3087 8730-0378=8352 8532-2358=6174 7641-1467=6174 *Therefore, we conclude that all T4 (with the exception of aaaa) converge to a fixed point 6174.

  6. T5 • Example: Start with the number 54321 54321-12345=41976 82962 75933 63954 61974 82962

  7. Conclusions for t1 through t10 • T1– Can’t do. • T2– repeated pattern; starts at 45 • T3– converges to fixed point 495 • T4– converges to fixed point 6174 • T5– repeated pattern; starts at 82962 • T6– repeated pattern; starts at 851742 • T7– repeated pattern; starts at 8429652 • T8– repeated pattern; starts at 75308643 • T9– repeated pattern; starts at 863098632 • T10– repeated pattern; starts at 8633086632

  8. Generalizations • Sum of the digits at the fixed point or at which the pattern starts to repeat follows another pattern: • T2: 45: 4+5=9 • T3: 495: 4+9+5=18 • T4: 6174: 6+1+7+4=18 • T5: 82962: 8+2+9+6+2=27 • T6: 851742: ….…………..=27 • T7: 8429652:….………….=36 • T8: 75308643:….………...=36 • T9: 863098632:…….…….=45 • T10: 8633086632:…..…….=45

  9. Recap • Tn converges to a fixed point for T3 and T4. For all other Tn (2 through 10) they have repeated patterns. • For different values, the number of steps to reach the fixed point or the repeated pattern varies. • For Tn, we can predict the sum of the digits using the following equations: When n is even: sum Tn= 9 (n/2) When n is odd: sum Tn= 9 ((n+1)/2) (applies for n >1)

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