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Q01 |
- Diagram : Fibonacci's sequence in Pascal triangle
- Q02 | - Fibonacci's sequence
- Q03 | - Find the 6th number of Fibonacci's sequence
- Q04 | - The recursion formula
- Q05 | - Compare Fibonacci's sequence and Lucas sequence
- Q02 | - Fibonacci's sequence
Q01. Diagram :
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Q02. Fibonacci's sequence
Sequences
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(1) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
(2) It is called rabbit population pattern
(3) Fibonacci's sequence and keppler definetion see MD2002 14 25
(4) Lucas sequence and fibonacci's squauence in ZL.txt
- T(0) = 0
- T(1) = 1
- T(n) = T(n - 2) + T(n - 1)
- T(2) = T(1) + T(0) = 1 + 0 = 1
- T(3) = T(2) + T(1) = 1 + 1 = 2
- T(4) = T(3) + T(2) = 2 + 1 = 3
- It is same as Fibonacci's sequence
- But T(0) = 2
- Each pair rabit will produce one pair after one month they are born
- There is no death of the pair
- 1st month : 1 = one pair A
- 2nd month : 1 = the original one pair A
- 3rd month : 2 = Pair A produce pair A1
- 4th month : 3 = A + A1 + pair A produce A2
- 5th month : 5 = A + A1 + A2 + pair A produce A3 + A1 produce A11
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Q03. Find the 6th number of Fibonacci's sequence
The number
- Use Pascal triangle as shown in Diagram we have 1 + 4 + 3 = 8
- Use recursion we have
- T(0) = 0
- T(1) = 1
- T(2) = T(1) + T(0) = 1 + 0 = 1
- T(3) = T(2) + T(1) = 1 + 1 = 2
- T(4) = T(3) + T(2) = 2 + 1 = 3
- T(5) = T(4) + T(3) = 3 + 2 = 5
- T(6) = T(4) + T(5) = 3 + 5 = 8
- The number is 4 + 10 + 6 + 1 = 21
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Q04. The recursion formula
Fibonacci's sequence
- T(0) = 0
- T(1) = 1
- T(n) = T(n-2) + T(n-1)
- T(0) = 2
- T(1) = 1
- T(n) = T(n-2) + T(n-1)
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Q05. Compare Fibonacci's sequence and Lucas sequence
- n .............. 0 .... 1 .... 2 .... 3 .... 4 .... 5 .... 6 ..... 7 ....
- Fibonacci's .... 0 .... 1 .... 1 .... 2 .... 3 .... 5 .... 8 .... 13 ....
- Lucas .......... 2 .... 1 .... 3 .... 4 .... 7 ... 11 ... 18 .... 29 ....
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