Question

1. use the pattern to answer the questions.

step 1
step 2
step 3
a. rate of change (slope): ______
b. how many circles are in step 6? ______
c. how many circles are in step 0? ______
a. rate of change (slope):
b. how many circles are in step 6?
c. how many circles are in step 0?

Explanation:

Step 1: Count circles in each step

• Step 1: 2 blue + 3 white = 5 circles.
• Step 2: 2 blue + 6 white = 8 circles.
• Step 3: 2 blue + 9 white = 11 circles.

Step 2: Find the rate of change (slope)

The number of circles forms a sequence: 5, 8, 11,... The difference between consecutive terms is \(8 - 5 = 3\) and \(11 - 8 = 3\). So the rate of change (slope) is 3.

Step 3: Find the formula for the number of circles

The sequence is arithmetic with first term \(a_1 = 5\) and common difference \(d = 3\). The formula for the \(n\)-th term of an arithmetic sequence is \(a_n = a_1 + (n - 1)d\). Substituting, we get \(a_n = 5 + (n - 1) \times 3 = 3n + 2\).

Step 4: Find the number of circles in Step 6

Using the formula \(a_n = 3n + 2\), for \(n = 6\): \(a_6 = 3(6) + 2 = 20\).

Step 5: Find the number of circles in Step 0

Using the formula \(a_n = 3n + 2\), for \(n = 0\): \(a_0 = 3(0) + 2 = 2\).

Answer:

s:
a. Rate of change (slope): \(\boldsymbol{3}\)
b. How many circles are in Step 6: \(\boldsymbol{20}\)
c. How many circles are in Step 0: \(\boldsymbol{2}\)