1 Chapter An Introduction to Problem Solving Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

1-2 Explorations with Patterns Students will be able to understand and explain • Finding patterns and determining if a pattern holds. • Deductive and inductive reasoning and when to use them. • Different types of sequences, such as arithmetic, geometric, and Fibonacci. • Finding the nth term of certain sequences. • Using differences to find a pattern for neither arithmetic nor geometric sequences.

EXAMPLE a. Describe any patterns: There are several possible patterns. For example, the numbers on the far left are natural numbers. The pattern starts with 1 and continues to the next greater natural number in each successive line.

EXAMPLE (continued) The numbers “in the middle” are products of two numbers, the second of which is 9. The first number in the first product is 0; after that the first number is formed using natural numbers and including one more in each successive line.

EXAMPLE (continued) The resulting numbers on the right are formed using 1s and include an additional 1 in each successive line.

EXAMPLE (continued) b. Do the patterns continue? Why or why not? The pattern in the complete equation appears to continue for a number of cases, but it does not continue in general; for example, This pattern breaks down when the pattern of digits in the number being multiplied by 9 contains previously used digits.

Inductive Reasoning Make generalizations based on observations and patterns. May lead to a conjecture – a statement that is thought to be true but hasn’t been proved. Can be proven false by a counterexample – a statement that contradicts the conjecture.

The Danger of Making Conjectures Choose points on a circle and connect them to form distinct, nonoverlapping regions. In this figure, 2 points determine 2 regions, 3 points determine 4 regions, and 4 points determine 8 regions. What is the maximum number of regions that would be determined by 10 points?

It appears that each time we increase the number of points by 1, we double the number of regions. If this were true, then for 5 points we would have 2 times the number of regions with 4 points, or and so on. If we base our conjecture on this pattern, we might believe that for 10 points, we would have 29, or 512 regions.

An initial check for this conjecture is to see whether we obtain 16 regions for 5 points.

For 6 points, the pattern predicts that the number of regions will be 32. There are actually 31 regions, so the conjecture is not true.

Definition Sequence: an ordered arrangement of numbers, figures, or objects. Sequences can be classified by their properties.

Arithmetic Sequences A sequence in which each successive term is obtained from the previous term by addition or subtraction of a fixed number (the difference). The sequence 2, 5, 8, 11, 14, … is an arithmetic sequence because the difference between each term is 3.

Example Find a pattern in the number of matchsticks required to continue the pattern. Note that the addition of a square to an arrangement requires the addition of three matchsticks each time. Thus, the numerical pattern obtained is 4, 7, 10, 13, 16, 19, … , an arithmetic sequence with a difference of 3.

Example Find the first four terms of a sequence given by and determine whether the sequence seems to be arithmetic. Number of Term Term 1 4·1 + 3 = 7 2 4·2 + 3 = 11 3 4·3 + 3 = 15 4 4·4 + 3 = 19 The sequence seems arithmetic with difference 4.

Example (continued) Find the first four terms of a sequence given by and determine whether the sequence seems to be arithmetic. Number of Term Term 1 12 − 1 = 0 2 22 − 1 = 3 3 32 − 1 = 8 4 42 − 1 = 15 The sequence is not arithmetic because it has no common difference.

Example The diagrams in the figure show the molecular structure of alkanes, a class of hydrocarbons. C represents a carbon atom and H a hydrogen atom. A connecting segment shows a chemical bond. (Remark: CH4 stands for C1H4.)

Example (continued) Hectane is an alkane with 100 carbon atoms. How many hydrogen atoms does it have? First study the drawing of the alkanes and disregard the extreme left and right hydrogen atoms in each. With this restriction, we see that for every carbon atom, there are two hydrogen atoms. Therefore, there are twice as many hydrogen atoms as carbon atoms plus the two hydrogen atoms at the extremes.

Example (continued) This can be summarized as follows.

Example A theater is set up in such a way that there are 20 seats in the first row and 4 additional seats in each consecutive row. The last row has 144 seats. How many rows are there in the theater? There are 32 rows in the theater.

Fibonacci Sequence The Fibonacci sequence is defined by for n = 3, 4, 5, … The sequence is: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

Geometric Sequences A sequence in which each successive term is obtained from the previous term by multiplying by a fixed nonzero number (the ratio). Example The sequence 2, 6, 18, 54, 162, … is a geometric sequence because the ratio between each term is 3.

Definition If n is a natural number, then n factors If n = 0 and a ≠ 0, then a0 = 1.

Other Sequences Triangular numbers Figurate numbers are sequences of numbers that can be represented by dots arranged in the shape of geometric figures. Square numbers

Example Use differences to find a pattern. Then, assuming that the pattern discovered continues, find the seventh term in the sequence. 5 6 14 29 51 80 1 8 15 22 29 7 7 7 7 The first difference is an arithmetic sequence with difference 7. The sixth term in the first difference row is 29 + 7 = 36, and the seventh term in the original sequence is 80 + 36 = 116.