Inductive Reasoning

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Show your work for each problem.

1.
Which of these is an example of inductive reasoning?
1. a)If x + 3 = 7, then x = 4.
2. b)All squares have 4 sides. This shape is a square, so it has 4 sides.
3. c)Every multiple of 10 ends in 0. 70 is a multiple of 10, so 70 ends in 0.
4. d)The bus was late every morning this week, so it will probably be late tomorrow.
2.
Use the conjecture "the sum of the first $n$ odd numbers is $n^2$ " to find $1 + 3 + 5 + 7 + 9 + 11$ without adding term by term.
3.
Dots are arranged in triangles: figure 1 has $1$ dot, figure 2 has $3$ dots, figure 3 has $6$ dots and figure 4 has $10$ dots. If the pattern continues, how many dots does figure 6 have?
4.
Which number is a counterexample to the conjecture "all prime numbers are odd"?
1. a)2
2. b)9
3. c)15
4. d)7
5.
Conjecture: "doubling a number always makes it bigger." Which number is equal to its own double and is therefore a counterexample?
6.
A sunflower is $2$ cm tall after week 1, $5$ cm after week 2, $8$ cm after week 3 and $11$ cm after week 4. If the pattern continues, how tall will it be after week 8 (in cm)?
7.
Conjecture: the sum of the first $n$ odd numbers is $n^2$ . Test it with $n = 5$ , then use it to predict the sum of the first $20$ odd numbers.
8.
Tom adds $1 + 5$ , $9 + 3$ and $15 + 7$ , gets an even number each time and concludes: "the sum of two odd numbers is always even." Which kind of reasoning did Tom use?
1. a)Circular reasoning
2. b)Mathematical induction
3. c)Inductive reasoning
4. d)Deductive reasoning
9.
A triangle has $0$ diagonals, a quadrilateral $2$ , a pentagon $5$ , a hexagon $9$ , a heptagon $14$ and an octagon $20$ . Continue the pattern: how many diagonals does a nonagon (9 sides) have?
10.
Look at these products: $11 \times 11 = 121$ , $111 \times 111 = 12321$ , $1111 \times 1111 = 1234321$ . Make a conjecture and predict $11111 \times 11111$ .
11.
Lena claims: "squaring a number always makes it bigger." Test her claim with $\frac{1}{2}$ : what is $\left(\frac{1}{2}\right)^2$ ? Write your answer as a fraction.
12.
The expression $n^2 + n + 41$ gives a prime number for every $n$ from $0$ to $39$ . For $n = 40$ it gives $1681$ . Which statement is correct?
1. a)The conjecture is proven because it worked for n = 0 to 39
2. b)One counterexample is not enough to reject the conjecture
3. c)1681 is prime, so the conjecture still holds
4. d)1681 = 41 × 41, so n = 40 is a counterexample
13.
At a party everyone shakes hands with everyone else exactly once. With $2$ people there is $1$ handshake, with $3$ people $3$ handshakes and with $4$ people $6$ handshakes. How many handshakes are there with $10$ people?
14.
Test the conjecture " $n^2 + n + 41$ is always prime" step by step and find the value that breaks it at $n = 40$ .
15.
From the cases $0, 2, 5, 9$ (polygons with $3$ to $6$ sides) we made the conjecture that a polygon with $n$ sides has $\frac{n(n - 3)}{2}$ diagonals. Use it: how many diagonals does a polygon with $12$ sides have?
16.
Sam notices: $2^2 - 1 = 3$ , $2^3 - 1 = 7$ , $2^5 - 1 = 31$ , $2^7 - 1 = 127$ , all prime. He conjectures: "if $p$ is prime, then $2^p - 1$ is prime." Which value of $p$ is a counterexample?
1. a)p = 11
2. b)p = 17
3. c)p = 13
4. d)p = 19
17.
Put points on a circle and join every pair of points with a straight line. Count the regions inside the circle: $1$ point gives $1$ region, $2$ points give $2$ , $3$ points give $4$ , $4$ points give $8$ , $5$ points give $16$ . A careful drawing with $6$ points, placed so that no three lines meet at one point, shows $31$ regions. Test the doubling conjecture: by how many regions does its prediction miss?
18.
Conjecture: "the sum of three consecutive whole numbers is always $3$ times the middle number." Test it inductively, then use the deductive version to find $49 + 50 + 51$ .