Question

1. sketch the next figure in the pattern.

images of three circles 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 ________.

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

1. conjecture: the product of any two odd numbers is ________.

1×1 = 1 9×11 = 99
1×3 = 3 13×11 = 143
3×7 = 21 101×103 = 10,403

Explanation:

Question 2 (Sketching the next figure)

Step 1: Analyze the rotation of the yellow sector

Observe the position of the yellow sector in each circle. The yellow sector rotates counterclockwise by one segment (each circle is divided into 6 segments) in each subsequent figure.

Step 2: Sketch the next figure

In the third figure, the yellow sector is at a certain position. Rotating it counterclockwise by one more segment (so it moves to the next adjacent segment counterclockwise) will give the next figure. The circle remains divided into 6 equal segments, and the yellow sector is now in the new rotated position.

Question 3 (Number sequence: 256, 64, 16, 4, ...)

Step 1: Identify the pattern

Check the relationship between consecutive terms. Divide each term by the previous term: $\frac{64}{256}=\frac{1}{4}$, $\frac{16}{64}=\frac{1}{4}$, $\frac{4}{16}=\frac{1}{4}$. So each term is $\frac{1}{4}$ of the previous term.

Step 2: Find the next term

To find the next term, multiply the last given term (4) by $\frac{1}{4}$. So $4\times\frac{1}{4} = 1$.

Step 1: Identify the pattern

Check the relationship between consecutive terms. Divide each term by the previous term: $\frac{6}{2}=3$, $\frac{18}{6}=3$, $\frac{54}{18}=3$. So each term is 3 times the previous term.

Step 2: Find the next term

To find the next term, multiply the last given term (54) by 3. So $54\times3=162$.

Answer:

1

Question 4 (Number sequence: 2, 6, 18, 54, ...)