Question

1. how can you prove that a conjecture is false?

1. 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, ... ______

1. 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 ________.

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 can find a counterexample. A counterexample is a specific case (an example) that satisfies the hypothesis of the conjecture but does not satisfy the conclusion. For example, if the conjecture is "All prime numbers are odd", the number 2 (which is prime) is a counterexample because it is prime but not odd, thus proving the conjecture false.

Looking at the pattern of the circles (divided into 6 equal parts with a yellow sector):

• First circle: Yellow sector is in the bottom - right (let's consider the sectors in order).
• Second circle: Yellow sector has rotated counter - clockwise by one sector.
• Third circle: Yellow sector has rotated counter - clockwise by another sector.

So, the next figure (fourth circle) should have the yellow sector rotated counter - clockwise by one more sector from the third circle's yellow sector position. The non - yellow sectors (the lines) also seem to have a rotational pattern, but the main pattern for the yellow sector is a counter - clockwise rotation of one sector per figure. So we sketch a circle with 6 equal sectors, and the yellow sector in the position obtained by rotating the third circle's yellow sector counter - clockwise by one sector.

Step 1: Identify the pattern

We check the relationship between consecutive terms. Let's divide each term by the previous term.
For \(256\) and \(64\): \(\frac{64}{256}=\frac{1}{4}\)
For \(64\) and \(16\): \(\frac{16}{64}=\frac{1}{4}\)
For \(16\) and \(4\): \(\frac{4}{16}=\frac{1}{4}\)
So the pattern is that each term is \(\frac{1}{4}\) of the previous term (or we can say we multiply the previous term by \(\frac{1}{4}\) to get the next term).

Step 2: Find the next term

To find the next term after \(4\), we multiply \(4\) by \(\frac{1}{4}\).
\(4\times\frac{1}{4} = 1\)

Answer:

By finding a counterexample (a specific case that satisfies the conjecture's hypothesis but not its conclusion).

Question 2 (Sketching the next figure)