2.1: Inductive Reasoning
Learning Objectives
- Recognize visual and number patterns.
- Write a counterexample.
Review Queue
- Look at the patterns of numbers below. Determine the next three numbers in the list.
- 1, 2, 3, 4, 5, 6, _____, _____, _____
- 3, 6, 9, 12, 15, _____, _____, _____
- 5, 1, -3, -7, -11, _____, _____, _____
- Are the statements below true or false? If they are false, state why.
- Perpendicular lines form four right angles.
- Linear pairs are always congruent.
- For the line, \begin{align*}y=3x+1\end{align*}:
- Find the slope.
- Find the \begin{align*}y-\end{align*}intercept.
- Make an \begin{align*}x-y\end{align*} table for \begin{align*}x = 1, 2, 3, 4,\end{align*} and 5.
Know What? This is the “famous” locker problem:
A new high school has just been completed. There are 100 lockers that are numbered 1 to 100. During recess, the students decide to try an experiment. The first student opens all of the locker doors. The second student closes all of the lockers with even numbers. The \begin{align*}3^{rd}\end{align*} student changes every \begin{align*}3^{rd}\end{align*} locker (change means closing lockers that are open, and opening lockers that are closed). The \begin{align*}4^{th}\end{align*} student changes every \begin{align*}4^{th}\end{align*} locker and so on.
Imagine that this continues until the 100 students have followed the pattern with the 100 lockers. At the end, which lockers will be open and which will be closed? Make a table to help you and use the following website:
http://www.mth.msu.edu/~nathsinc/java/Lockers/
Visual Patterns
Inductive Reasoning: Making conclusions based upon examples and patterns.
Let’s look at some patterns to get a feel for what inductive reasoning is.
Example 1: A dot pattern is shown below. How many dots would there be in the \begin{align*}4^{th}\end{align*} figure? How many dots would be in the \begin{align*}6^{th}\end{align*} figure?
Solution: Draw a picture. Counting the dots, there are \begin{align*}4 + 3 + 2 + 1 = 10 \ dots\end{align*}.
For the \begin{align*}6^{th}\end{align*} figure, we can use the same pattern, \begin{align*}6 + 5 + 4 + 3 + 2 + 1\end{align*}. There are 21 dots in the \begin{align*}6^{th}\end{align*} figure.
Example 2: How many triangles would be in the \begin{align*}10^{th}\end{align*} figure?
Solution: There would be 10 squares in the \begin{align*}10^{th}\end{align*} figure, with a triangle above and below each one. There is also a triangle on each end of the figure. That makes \begin{align*}10 +10 + 2 = 22\end{align*} triangles in all.
Example 3: For two points, there is one line segment between them. For three non-collinear points, there are three segments. For four points, how many line segments are between them? If you add a fifth point, how many line segments are between the five points?
Solution: Draw a picture of each and count the segments.
For 4 points there are 6 line segments and for 5 points there are 10 line segments.
Number Patterns
Let’s look at a few examples.
Example 4: Look at the pattern 2, 4, 6, 8, 10, \begin{align*}\ldots\end{align*} What is the \begin{align*}19^{th}\end{align*} term in the pattern?
Solution: For part a, each term is 2 more than the previous term.
You could count out the pattern until the \begin{align*}19^{th}\end{align*} term, but that could take a while. Notice that the \begin{align*}1^{st}\end{align*} term is \begin{align*}2 \cdot 1\end{align*}, the \begin{align*}2^{nd}\end{align*} term is \begin{align*}2 \cdot 2\end{align*}, the \begin{align*}3^{rd}\end{align*} term is \begin{align*}2 \cdot 3\end{align*}, and so on. So, the \begin{align*}19^{th}\end{align*} term would be \begin{align*}2 \cdot 19\end{align*} or 38.
Example 5: Look at the pattern 1, 3, 5, 7, 9, 11, \begin{align*}\ldots\end{align*} What is the \begin{align*}34^{th}\end{align*} term in the pattern?
Solution: The next term would be 13 and continue go up by 2. Comparing this pattern to Example 4, each term is one less. So, we can reason that the \begin{align*}34^{th}\end{align*} term would be \begin{align*}34 \cdot 2\end{align*} minus 1, which is 67.
Example 6: Look at the pattern: 3, 6, 12, 24, 48, \begin{align*}\ldots\end{align*}
a) What is the next term in the pattern?
b) The \begin{align*}10^{th}\end{align*} term?
Solution: This pattern is different than the previous two examples. Here, each term is multiplied by 2 to get the next term.
Therefore, the next term will be \begin{align*}48 \cdot 2\end{align*} or 96. To find the \begin{align*}10^{th}\end{align*} term, continue to multiply by 2, or \begin{align*}3 \cdot \underbrace{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}_{2^9} = 1536\end{align*}.
Example 7: Find the \begin{align*}8^{th}\end{align*} term in the list of numbers: \begin{align*}2,\frac{3}{4},\frac{4}{9},\frac{5}{16},\frac{6}{25}\ldots\end{align*}
Solution: First, change 2 into a fraction, or \begin{align*}\frac{2}{1}\end{align*}. So, the pattern is now \begin{align*}\frac{2}{1},\frac{3}{4},\frac{4}{9},\frac{5}{16},\frac{6}{25}\ldots\end{align*} The top is 2, 3, 4, 5, 6. It increases by 1 each time, so the \begin{align*}8^{th}\end{align*} term’s numerator is 9. The denominators are the square numbers, so the \begin{align*}8^{th}\end{align*} term’s denominator is \begin{align*}8^2\end{align*} or 64. The \begin{align*}8^{th}\end{align*} term is \begin{align*}\frac{9}{64}\end{align*}.
Conjectures and Counterexamples
Conjecture: An “educated guess” that is based on examples in a pattern.
Example 8: Here’s an algebraic equation and a table of values for \begin{align*}n\end{align*} and the result, \begin{align*}t\end{align*}.
\begin{align*}t=(n-1)(n-2)(n-3)\end{align*}
\begin{align*}n\end{align*} | \begin{align*}(n-1)(n-2)(n-3)\end{align*} | \begin{align*}t\end{align*} |
---|---|---|
1 | \begin{align*}(0)(-1)(-2)\end{align*} | 0 |
2 | \begin{align*}(1)(0)(-1)\end{align*} | 0 |
3 | \begin{align*}(2)(1)(0)\end{align*} | 0 |
After looking at the table, Pablo makes this conjecture:
The value of \begin{align*}(n-1)(n-2)(n-3)\end{align*} is 0 for any number \begin{align*}n\end{align*}.
Is this a true conjecture?
Solution: This is not a valid conjecture. If Pablo were to continue the table to \begin{align*}n = 4\end{align*}, he would have see that \begin{align*}(n-1)(n-2)(n-3)=(4-1)(4-2)(4-3)=(3)(2)(1)=6\end{align*}
In this example \begin{align*}n = 4\end{align*} is called a counterexample.
Counterexample: An example that disproves a conjecture.
Example 9: Arthur is making figures for an art project. He drew polygons and some of their diagonals.
From these examples, Arthur made this conjecture:
If a convex polygon has \begin{align*}n\end{align*} sides, then there are \begin{align*}n - 3\end{align*} triangles drawn from any vertex of the polygon.
Is Arthur’s conjecture correct? Or, can you find a counterexample?
Solution: The conjecture appears to be correct. If Arthur draws other polygons, in every case he will be able to draw \begin{align*}n - 3\end{align*} triangles if the polygon has \begin{align*}n\end{align*} sides.
Notice that we have not proved Arthur’s conjecture, but only found several examples that hold true. So, at this point, we say that the conjecture is true.
Know What? Revisited The table below is the start of the 100 lockers and students. Students are vertical and the lockers are horizontal. \begin{align*}X\end{align*} means the locker is closed, \begin{align*}O\end{align*} means the locker is open.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} |
2 | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | ||||||||
3 | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | \begin{align*}X\end{align*} | |||||||||||
4 | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | ||||||||||||
5 | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | \begin{align*}O\end{align*} | |||||||||||||
6 | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | ||||||||||||||
7 | \begin{align*}X\end{align*} | \begin{align*}O\end{align*} | ||||||||||||||
8 | \begin{align*}X\end{align*} | \begin{align*}X\end{align*} | ||||||||||||||
9 | \begin{align*}O\end{align*} | |||||||||||||||
10 | \begin{align*}X\end{align*} | |||||||||||||||
11 | \begin{align*}X\end{align*} | |||||||||||||||
12 | \begin{align*}X\end{align*} | |||||||||||||||
13 | \begin{align*}X\end{align*} | |||||||||||||||
14 | \begin{align*}X\end{align*} | |||||||||||||||
15 | \begin{align*}X\end{align*} | |||||||||||||||
16 | \begin{align*}O\end{align*} |
If you continue on in this way, the numbers that follow the pattern: 1, 4, 9, 16, \begin{align*}\ldots\end{align*} are going to be the only open lockers. These numbers are called square numbers and they are: 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
Review Questions
- Questions 1-5 are similar to Examples 1, 2a, and 3.
- Questions 6-17 are similar to Examples 4-7.
- Questions 18-25 are similar to Examples 8 and 9.
For questions 1-3, determine how many dots there would be in the \begin{align*}4^{th}\end{align*} and the \begin{align*}10^{th}\end{align*} pattern of each figure below.
- Use the pattern below to answer the questions.
- Draw the next figure in the pattern.
- How does the number of points in each star relate to the figure number?
- Use the pattern below to answer the questions. All the triangles are equilateral triangles.
- Draw the next figure in the pattern. How many triangles does it have?
- Determine how many triangles are in the \begin{align*}24^{th}\end{align*} figure.
For questions 6-13, determine: the next three terms in the pattern.
- 5, 8, 11, 14, 17, \begin{align*}\ldots\end{align*}
- 6, 1, -4, -9, -14, \begin{align*}\ldots\end{align*}
- 2, 4, 8, 16, 32, \begin{align*}\ldots\end{align*}
- 67, 56, 45, 34, 23, \begin{align*}\ldots\end{align*}
- 9, -4, 6, -8, 3, \begin{align*}\ldots\end{align*}
- \begin{align*}\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6} \ldots\end{align*}
- \begin{align*}\frac{2}{3},\frac{4}{7},\frac{6}{11},\frac{8}{15},\frac{10}{19}, \ldots\end{align*}
- -1, 5, -9, 13, -17, \begin{align*}\ldots\end{align*}
For questions 14-17, determine the next two terms and describe the pattern.
- 3, 6, 11, 18, 27, \begin{align*}\ldots\end{align*}
- 3, 8, 15, 24, 35, \begin{align*}\ldots\end{align*}
- 1, 8, 27, 64, 125, \begin{align*}\ldots\end{align*}
- 1, 1, 2, 3, 5, \begin{align*}\ldots\end{align*}
For questions 18-23, give a counterexample for each of the following statements.
- If \begin{align*}n\end{align*} is a whole number, then \begin{align*}n^2 > n\end{align*}.
- Every prime number is an odd number.
- All numbers that end in 1 are prime numbers.
- All positive fractions are between 0 and 1.
- Any three points that are coplanar are also collinear.
- Congruent supplementary angles are also linear pairs.
Use the following story for questions 24 and 25.
A car salesman sold 5 used cars to five different couples. He noticed that each couple was under 30 years old. The following day, he sold a new, luxury car to a couple in their 60’s. The salesman determined that only younger couples by used cars.
- Is the salesman’s conjecture logical? Why or why not?
- Can you think of a counterexample?
Review Queue Answers
- 7, 8, 9
- 18, 21, 24
- 36, 49, 64
- true
- false,
- \begin{align*}m = 3\end{align*}
- \begin{align*}b = 1\end{align*}
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
1 | 4 |
2 | 7 |
3 | 10 |
4 | 13 |
5 | 16 |