Question

practice use the sequence and inductive reasoning to make a conjecture. 3. what is the color of the thirtieth figure? 4. what is the shape of the fortieth figure?

Explanation:

Step1: Identify color cycle length

The color sequence repeats every 4 figures: light pink, red, dark gray, light pink, red, gray, light pink, red... So cycle length $n_c=4$.

Step2: Calculate position in color cycle

For the 30th figure: $30 \div 4 = 7$ with a remainder of $2$.

Step3: Match remainder to color

The 2nd position in the color cycle is red.

Step4: Identify shape cycle length

The shape sequence repeats every 4 figures: circle, triangle, square, star, circle, triangle, square, star... So cycle length $n_s=4$.

Step5: Calculate position in shape cycle

For the 40th figure: $40 \div 4 = 10$ with a remainder of $0$. A remainder of 0 means it matches the 4th position in the cycle.

Step6: Match remainder to shape

The 4th position in the shape cycle is star.

Answer:

1. The color of the thirtieth figure is red.
2. The shape of the fortieth figure is star.