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- 1. Fibonacci Sequence ROSANNE U.DIADULA Course Instructor RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY SECOND SEMESTER A.Y. 2024 - 2025 F IB O N A C C I S EQ U E N C E
- 2. FIBONACCI SEQUENCE DEFINITION The FibonacciSequence is a series of numbers where each term is the sum of the two preceding ones. • Defined as 𝐹1 = 1, 𝐹2 = 1 • 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 for 𝑛 ≥ 3 The sequence starts at 1, 1, 2, 3, 5, 8, 13, … RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E
- 3. FIBONACCI SEQUENCE DEFINITION The FibonacciSequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,… RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E The next number is found by adding up the two numbers before it: the 2 is found by adding the two numbers before it (1 + 1), the 3 is found by adding the two numbers before it (1 + 2), the 5 is (2 + 3), and so on!
- 4. FIBONACCI SEQUENCE DEFINITION RO SANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E 𝒏 = 0 1 2 3 4 5 6 7 8 9 10 11 𝟏𝟐 𝟏𝟑 𝟏𝟒 … 𝐹𝑛 0 1 1 2 3 5 8 13 21 34 55 89 11 233 377 … The rule is 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 where 𝐹𝑛 is term number ‘n’ 𝐹𝑛−1 is the previous term (𝑛 − 1) 𝐹𝑛−2 is the term before that (𝑛 − 2)
- 5. FIBONACCI SEQUENCE DEFINITION RO SANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E 𝒏 = 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 … 𝐹𝑛 0 1 1 2 3 5 8 13 21 34 55 89 11 233 377 … 𝑨 𝑩 𝑩 𝑨 2 3 1.5 3 5 1.6666666 … 5 8 1.6 8 13 1.625 … … … 144 233 1.618055556 … 233 377 1.618025751 … … … … When we take any two successive Fibonacci numbers, their ratio is very close to the Golden Ratio “𝜑” which is approximately 1.618034…
- 6. USING THE GOLDENRATIO TO CALCULATE FIBONACCI NUMBERS Formula 𝐹𝑛 = 𝜑𝑛 − 1 − 𝜑 𝑛 5 where 𝜑 ≈ 1.618034… RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E
- 7. USING BINET’S FORMULATO CALCULATE FIBONACCI NUMBERS Formula 𝐹𝑛 = 1 + 5 𝑛 − 1 − 5 𝑛 2𝑛 5 RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E
- 8. FIBONACCI SEQUENCE RO SANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E EXAMPLES: 1. Find 𝐹6. 𝐹𝑛 = 𝜑𝑛 − 1 − 𝜑 𝑛 5 𝐹6 = (1.618034)6 − 1 − 1.618034 6 5 𝐹6 = 8.00000033
- 9. FIBONACCI SEQUENCE RO SANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E EXAMPLES: 2. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑒𝑥𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑎𝑓𝑡𝑒𝑟 8? 𝐼𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 8 𝑡𝑖𝑚𝑒𝑠 𝜑. 8𝜑 = 8 × 1.618034 … = 12.94427 … = 13
- 10. FIBONACCI SEQUENCE RO SANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E EXAMPLES: 2. 𝑊ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑛𝑒𝑥𝑡 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒 𝑎𝑓𝑡𝑒𝑟 8? 𝐼𝑡 𝑤𝑖𝑙𝑙 𝑏𝑒 8 𝑡𝑖𝑚𝑒𝑠 𝜑. 8𝜑 = 8 × 1.618034 … = 12.94427 … = 13
- 11. QUIZ Answer completely. 1. Let𝐹𝑛 be the nth term of the Fibonacci sequence with 𝐹1 = 1, 𝐹2 = 1, 𝐹3 = 2, and so on. a. Find 𝐹10, 𝐹23, 𝑎𝑛𝑑 𝐹30. RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E 2. If 𝐹22 = 17,711 and 𝐹24 = 46,368, what is 𝐹23?
- 12. IMPORTANT NOTICE For ClassroomUse Only This material is intended for classroom use only. Please do not reproduce without permission. RO S ANNE U . DI ADUL A M110: NUMBER TH EO RY F IB O N A C C I S EQ U E N C E