CHAPTER 1 Nature of Mathematics Rationale Learning Objectives: 1. Know the theories and concepts of nature of mathematics 2. Be able to identify different formulas In patterns and sequence 3. To familiarize with the symbols, formula and establish rules in each topic Learning Outcomes: 1. Develop the mathematical analysis to understand sequence and patterns. 2. Recognize patterns and appreciate the Nature of Mathematics in the Modern World PATTERN A pattern is a series or sequence that repeats. You can observe patterns - things like colors, shapes, actions, or other sequences that repeat - everywhere. Think about words or melodies in songs, lines and curves on buildings, or even in the grocery store where boxes and jars of various items are lined up. But, one of the most common places to find patterns is in math. Math patterns are sequences that repeat according to a rule or rules. Example: 1. Find the pattern. 1, 4, 7, 10, 13, 16, _? Answer: starts at 1 and jumps 3 every time. 19 2. Find the pattern. The rectangular numbers are the next sequence in the progression of patterns. Instead of the expression (or ) as with square numbers, the rectangular numbers use for one dimension. EXPONENTIAL GROWTH MODEL In exponential growth, a population's per capita (per individual) growth rate stays the same regardless of population size, making the population grow faster and faster as it gets larger. In nature, populations may grow exponentially for some period, but they will ultimately be limited by resource availability. The formula is A=Pert where: A= the size of the population after it grows P= the initial number of people r= rate of growth t= time e= 2.718 Example: A=1,236,000; r=12%, i=50 years, P=? Find the population number. Answer: Since the formula is A=Pert , transposition would be an appropriate step to get the initial number of people P=Aert P= 1,236,000 (2.718(12%)(50)) P=1,236,000 (2.718(6)) P=1,236,000 (403.1778962) P=498,327,880 Sequence A list of numbers or objects in a special order. It is a particular order in which related things follow each other. Example: 3, 5, 7, 9, ... is a sequence starting at 3 and increasing by 2 each time. Fibonacci Sequence The Fibonacci sequence is a set of numbers that starts with a one or a zero, followed by a one, and proceeds based on the rule that each number (called a Fibonacci number) is equal to the sum of the preceding two numbers. If the Fibonacci sequence is denoted F (n), where n is the first term in the sequence, the following equation obtains for n = 0, where the first two terms are defined as 0 and 1 by convention. Example: F (0) = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ... Patterns 1. 2. 2. What letter comes next in this pattern? ZEF YGH XIJ WKL _____ 3. What completes the following pattern? 50, 55, 65, 80, 100, ___ Fibonacci sequence Find F(39). 1 1+√5 1−√5 2 2 F(39) = √5 [( )]39 − [( )]39 F(39) =63,245,986 If Fib(18) = 2,854 and Fib (19) = 4, 181 What is Fib (20) and Fib (21) F(20) = 1 1+√5 1−√5 2 2 [( √5 F(20) = 6,765 )]20 − [( )]20 1 F(21) = √5 [( 1+√5 1−√5 2 2 F(21) = 10,946 )]21 − [( )]21 Evaluate the following: a. Fib(31) + Fib(32) = b. Fib (15) + Fib(16) +Fib(17)= Suppose the population in a city is 65,000. The population size is given by the exponential growth model A= 65,000𝑒 0.65 . Find t when the population would be