Fibonacci Numbers & The Golden Ratio PDF

Fibonacci numbers & the Golden Ratio GE 1108 MARIAN B. SANTOS MS, RN LPT  Mathematics is not just solving for X But also figuring out whY… Recall: Learning Objectives: At the end of the lesson, learners will be able to: 1. Articulate and solve sequence...

Fibonacci numbers & the Golden Ratio GE 1108 MARIAN B. SANTOS MS, RN LPT  Mathematics is not just solving for X But also figuring out whY… Recall: Learning Objectives: At the end of the lesson, learners will be able to: 1. Articulate and solve sequences and recurrences relation. 2. Enumerate the contribution of Leonardo Pisano in Mathematics. 3. Explain Fibonacci Numbers and apply Binet’s formula to solve the nth term of a given Fibonacci sequence. 4. Describe the relationship of Fibonacci sequence and the Golden Ratio. Sequences and Recurrence Relation Terms to remember: Pattern: repeated arrangement of numbers, shapes, colors, & others. Can be related to any type of event or object the set of numbers are related to each other in a specific rule Finite or infinite Known as a sequence Sequences and Recurrence Relation A sequence is a function from a subset of integer ℤ to a set 𝑆. We use the notation 𝑎𝑛 to denote the image of the integer 𝑛. We call 𝑎𝑛 a term of the sequence. 1 Ex. Write the first five terms of the sequence 𝑏𝑛 = 2𝑛 Defining sequence using previous terms A recurrence relation is an equation that defines a sequence based on a rule that gives the next term as a function of the previous term(s). The simplest form of a recurrence relation is the case where the next term depends only on the immediately previous term. Example2: Define a sequence recursively by 𝑎1 = 1, 𝑎2 = 3, 𝑎3 = 10, and the rule that 𝑎𝑛 = 𝑎𝑛−1 − 𝑎𝑛−3. Compute 𝑎5 Arithmetic Sequences An arithmetic progression (sometimes called an arithmetic sequence) is a sequence where each term differs from the next by the same, fixed quantity. Example 3: Write the first five terms of the sequence given by the rule 𝑎𝑛 =6 + 4𝑛 Arithmetic Sequences General formula: 𝑎1 = 𝑚 𝑛 − 1 + 𝑎1 Example 3: Consider the sequence 3, 8, 13,18, 23, 28... Find 𝑎101 Geometric Progression A geometric progression (sometimes called a geometric sequence) is a sequence where the ratio between subsequent terms is the same, fixed quantity. Example 4: 𝑎1 = 10, 𝑎2 = 30, 𝑎3 = 90, 𝑎4 = 270,... and given by the rule 𝑎𝑛 = 10 ∙ 3𝑛−1 Think – Pair - Share Identify the next element for each set and state the rule. 1. 2, 4, 6, 8, ___ 2. 11, 20, 29, 38, 47, ___ 3. 42, 40, 38, 35, 33, 31, 28, ___ 4. F2, E4, D8, C16, ___ Fibonacci Numbers Leonardo Pisano ▪ Born in Pisa, Italy in 1175 AD ▪ Grew up with a North African education under the Moors ▪ Traveled extensively around the Mediterranean coast ▪ Met with many merchants and learned their systems of arithmetic ▪ Realized the advantages of the Hindu-Arabic system Fibonacci’s Mathematical Contributions Introduced the Hindu- Arabic number system into Europe. Based on ten digits and a decimal point. Europe previously used the Roman number system. Consisted of Roman numerals. Persuaded mathematicians to use the Hindu-Arabic number system The Fibonacci Rabbit Problem The Fibonacci Rabbit Problem Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits can mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was... How many pairs will there be in one year? Generalize Exponential Growth The Fibonacci Numbers Were introduced in The Book of Calculating (Abacus) Originally the series begins with 0 and 1 Later on, Fibonacci omitted the 0 and starts and 1 Next number is found by adding the last two numbers together Number obtained is the next number in the series Pattern is repeated over and over 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … 𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 The 16th term of the Fibonacci sequence is 987 and the 17th term is 1597. What is the 19th term? What is the 25th Fibonacci number? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, … What is the 25th Fibonacci number? Fibonacci numbers in nature How does Fibonacci numbers is related to Golden Ratio? What is a Golden Ratio? Two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Golden Ratio representation Golden Rectangle Is a rectangle in which the ratio of the longer side to the shorter side is the golden ratio. The Golden Spiral Is a logarithmic spiral whose growth factor is 𝜑, the golden ratio. The spiral gets wider by a factor of 𝜑 every quarter turn. The Golden Spiral The Golden Ratio The Golden Ratio and The Fibonacci Numbers Golden Spiral/Ratio in Cinematography  Seatwork: WHAT IS THE ESSENCE OF LEARNING FIBONACCI NUMBERS AND THE GOLDEN RATIO? Kamsahamnida!