Question

name: mirk...
patterns & conjectures

1. how can you prove that a conjecture is false?
2. sketch the next figure in the pattern. three circle diagrams with yellow sectors

describe a pattern in the sequence of numbers. predict the next number.

1. 256, 64, 16, 4, ...
2. 2, 6, 18, 54, ...

complete the conjecture based on the pattern you observe in the examples below.

1. conjecture: the sum of any two odd numbers is ______.

examples: 1+1=2; 9+11=20; 1+5=6; 13+21=34; 7+9=16; 101+103=204

Explanation:

Question 1

To prove a conjecture is false, we find a counterexample. A counterexample is a specific case that satisfies the conjecture's hypothesis but not its conclusion. For example, if a conjecture says "all even numbers are divisible by 4", the number 2 (even) is not divisible by 4, so it's a counterexample disproving the conjecture.

Looking at the three given circle (divided into 8 equal parts) figures:

• First circle: yellow sector at the bottom - right (let's say position 1, if we number sectors from top - right clockwise as 1 - 8).
• Second circle: yellow sector at the bottom - left (position 2).
• Third circle: yellow sector at the left - middle (position 3).

The pattern of the yellow sector's position: it moves one sector counter - clockwise each time. So the next (fourth) figure should have the yellow sector at position 4 (left - top), moving counter - clockwise from the third figure's yellow sector position.

Step 1: Identify the pattern

We check the relationship between consecutive terms. Let's divide each term by the next term: $\frac{256}{64} = 4$, $\frac{64}{16}=4$, $\frac{16}{4} = 4$. So each term is divided by 4 to get the next term.

Step 2: Find the next term

To find the next term after 4, we divide 4 by 4. So $4\div4=1$.

Answer:

By finding a counterexample (a specific case where the conjecture's hypothesis is true but the conclusion is false).

Question 2 (Sketching the next figure)