Question

inductive reasoning and conjecture
make a conjecture about the next item in each sequence then find the next item in the sequence.

1. 5, 14, 13, 22
2. 2, 1, 1/2, 1/4, 1/8
3. 12, 6, 3, 1.5, 0.75

make a conjecture about each value or geometric relationship. (a picture may help you see a conjecture but a picture alone is not a conjecture.)

1. ∠abc is a right - angle
2. point t is between r and 7
3. bd bisects ∠abc
4. abcd is a square
5. ramps

rodney is rolling marbles down a ramp. every second that passes, he measures how far the marbles travel. he records the information in the table shown below.
a. make a conjecture about how far the marble travels as time increases.

second	1st	2nd	3rd	4th
distance (cm)	20	80	180	320

b. what will the distance the marble traveled be at the 6th second?

1. genealogy

miranda is developing a chart that shows her ancestry. she makes the three sketches shown below. the first dot represents herself. the second sketch represents herself and her parents. the third sketch represents herself, her parents, and her grandparents.
sketch what you think would be the next figure in the sequence.

Explanation:

Step1: Analyze sequence 1

The sequence is 5, 14, 23, 32. The common - difference is \(14 - 5=9\), \(23 - 14 = 9\), \(32 - 23=9\). So the next term is \(32+9 = 41\).

Step2: Analyze sequence 2

The sequence is \(2,1,\frac{1}{2},\frac{1}{4}\). The common - ratio is \(\frac{1}{2}\div1=\frac{1}{2}\), \(1\div2=\frac{1}{2}\), \(\frac{1}{4}\div\frac{1}{2}=\frac{1}{2}\). So the next term is \(\frac{1}{4}\times\frac{1}{2}=\frac{1}{8}\).

Step3: Analyze sequence 3

The sequence is 12, 6, 3, 1.5, 0.75. The common - ratio is \(6\div12=\frac{1}{2}\), \(3\div6=\frac{1}{2}\), \(1.5\div3=\frac{1}{2}\), \(0.75\div1.5=\frac{1}{2}\). So the next term is \(0.75\times\frac{1}{2}=0.375\).

Step4: Analyze problem 8a

The distances are 20, 80, 180, 320. The differences between consecutive terms are \(80 - 20 = 60\), \(180 - 80=100\), \(320 - 180 = 140\). The second - differences are \(100 - 60=40\), \(140 - 100 = 40\). The formula for the \(n\)th term of a quadratic sequence is \(an^{2}+bn + c\). Substituting \(n = 1\), \(a + b + c=20\); \(n = 2\), \(4a+2b + c=80\); \(n = 3\), \(9a+3b + c=180\). Solving the system of equations: \((4a + 2b + c)-(a + b + c)=80 - 20\) gives \(3a + b=60\), \((9a+3b + c)-(4a + 2b + c)=180 - 80\) gives \(5a + b = 100\). Then \((5a + b)-(3a + b)=100 - 60\), \(2a=40\), \(a = 20\), \(b = 0\), \(c = 0\). The conjecture is that the distance \(d\) in centimeters as a function of time \(t\) (in seconds) is \(d = 20t^{2}\).

Step5: Analyze problem 8b

Using the formula \(d = 20t^{2}\), when \(t = 6\), \(d=20\times6^{2}=20\times36 = 720\) cm.

Answer:

1. 41
2. \(\frac{1}{8}\)
3. 0.375

8a. The distance the marble travels is given by the formula \(d = 20t^{2}\) (where \(d\) is in cm and \(t\) is in seconds).
8b. 720 cm