Question

write a conjecture that describes the pattern in each sequence. then use your conjecture to find the next term in the sequence.

1. 4, 8, 12, 16, 20
2. 2, 22, 222, 2222
3. 1, 1/2, 1/4, 1/8
4. 6, 11/2, 5, 9/2, 4
5. arrival times: 3:00 p.m., 12:30 p.m., 10:00 a.m.,...
6. percent humidity: 100%, 93%, 86%,...

7.
8.
examples 2 and 3
make a conjecture about each value or geometric relationship.

1. the product of two odd numbers
2. the product of two and a number, plus one
3. the relationship between a and c if ab = bc, b ≠ 0
4. the relationship between a and b if ab = 1
5. the relationship between two intersecting lines that form four congruent angles
6. the relationship between the angles of a triangle with all sides congruent
7. the relationship between np and pq if point p is the midpoint of nq
8. the relationship between the volume of a prism and a pyramid with the same base and equal heights

Explanation:

Step1: Analyze sequence 1

The sequence 4, 8, 12, 16, 20 has a common - difference of 4. Each term is obtained by adding 4 to the previous term. The next term is \(20 + 4=24\).

Step2: Analyze sequence 2

The sequence 2, 22, 222, 2222 is formed by adding a digit 2 at the end of the previous number. The next term is 22222.

Step3: Analyze sequence 3

The sequence \(1,\frac{1}{2},\frac{1}{4},\frac{1}{8}\) is a geometric sequence with a common ratio of \(\frac{1}{2}\). The next term is \(\frac{1}{8}\times\frac{1}{2}=\frac{1}{16}\).

Step4: Analyze sequence 4

The sequence \(6,\frac{11}{2},5,\frac{9}{2},4\) can be rewritten as \(\frac{12}{2},\frac{11}{2},\frac{10}{2},\frac{9}{2},\frac{8}{2}\). The numerators decrease by 1 each time. The next term is \(\frac{7}{2}\).

Step5: Analyze arrival - times sequence

The time differences are 2.5 hours. 3:00 P.M. to 12:30 P.M. is a 2.5 - hour decrease, and 12:30 P.M. to 10:00 A.M. is a 2.5 - hour decrease. The next arrival time is 7:30 A.M.

Step6: Analyze humidity sequence

The sequence 100%, 93%, 86% has a common - difference of 7%. The next term is \(86 - 7=79\%\).

Step7: Analyze circle - shading sequence

The shaded part of the circle moves one - quarter turn clockwise each time. The next figure will have the shaded part in the top - right quadrant.

Step8: Analyze rectangle - grid sequence

The number of small rectangles in each figure is decreasing. The first has 8, the second has 6, the third has 4. The next figure will have 2 small rectangles.

Step9: Analyze product of two odd numbers

Let the two odd numbers be \(2m + 1\) and \(2n+1\), where \(m\) and \(n\) are integers. \((2m + 1)(2n + 1)=4mn+2m + 2n+1=2(2mn + m + n)+1\), which is odd. Conjecture: The product of two odd numbers is odd.

Step10: Analyze product of two and a number plus one

Let the number be \(x\). The expression is \(2x + 1\). Conjecture: The result is an odd number.

Step11: Analyze \(ab = bc\) with \(b

eq0\)
Dividing both sides of the equation \(ab = bc\) by \(b\) (since \(b
eq0\)), we get \(a = c\). Conjecture: If \(ab = bc\) and \(b
eq0\), then \(a = c\).

Step12: Analyze \(ab = 1\)

If \(ab = 1\), then \(a=\frac{1}{b}\) and \(b=\frac{1}{a}\). Conjecture: \(a\) and \(b\) are reciprocals of each other.

Step13: Analyze intersecting lines forming four congruent angles

If two intersecting lines form four congruent angles, then each angle is \(90^{\circ}\). Conjecture: The two lines are perpendicular.

Step14: Analyze angles of an equilateral triangle

In a triangle with all sides congruent (equilateral triangle), all angles are congruent. Since the sum of the interior angles of a triangle is \(180^{\circ}\), each angle is \(60^{\circ}\). Conjecture: The angles of an equilateral triangle are all \(60^{\circ}\).

Step15: Analyze mid - point of a line segment

If \(P\) is the mid - point of \(\overline{NQ}\), then \(NP=PQ\). Conjecture: The two line segments \(NP\) and \(PQ\) are congruent.

Step16: Analyze volume of prism and pyramid

The volume of a prism with base area \(B\) and height \(h\) is \(V_p=Bh\), and the volume of a pyramid with base area \(B\) and height \(h\) is \(V_y=\frac{1}{3}Bh\). Conjecture: The volume of a prism with the same base and equal height as a pyramid is three times the volume of the pyramid.

Answer:

1. 24
2. 22222
3. \(\frac{1}{16}\)
4. \(\frac{7}{2}\)
5. 7:30 A.M.
6. 79%
7. Shaded part in top - right quadrant
8. Figure with 2 small rectangles
9. The product of two odd numbers is odd.
10. The result is an odd number.
11. If \(ab = bc\) and \(b

eq0\), then \(a = c\).

1. \(a\) and \(b\) are reciprocals of each other.
2. The two lines are perpendicular.
3. The angles of an equilateral triangle are all \(60^{\circ}\).
4. \(NP = PQ\)
5. The volume of a prism with the same base and equal height as a pyramid is three times the volume of the pyramid.