Question

1. here is a pattern of growing rectangles: a. describe how the rectangle grows from step 0 to step 2. b. write an equation for sequence s, so that s(n) is the number of squares in step n. c. is s a geometric sequence, an arithmetic sequence, or neither? explain how you know. (from unit 1, lesson 10.)

Explanation:

Step1: Analyze rectangle growth

Count the number of squares in each step - by - step rectangle. In step 0, assume it starts with some base number of squares. Observe that the rectangle grows by adding a certain number of squares in each subsequent step.

Step2: Determine growth pattern

Let's assume the number of squares in step 0 is \(a_0\). If we find that the difference between the number of squares in consecutive steps is a constant \(d\), we can write the formula for the number of squares \(S(n)\) in step \(n\) as an arithmetic - sequence formula \(S(n)=a_0 + nd\). If the ratio between the number of squares in consecutive steps is a constant \(r\), we can write it as a geometric - sequence formula \(S(n)=a_0r^n\).

Step3: Check for arithmetic or geometric

To check if it's arithmetic, calculate \(S(1)-S(0)\), \(S(2)-S(1)\), etc. If these differences are equal, it's an arithmetic sequence. To check if it's geometric, calculate \(\frac{S(1)}{S(0)}\), \(\frac{S(2)}{S(1)}\), etc. If these ratios are equal, it's a geometric sequence.

Answer:

a. Without seeing the actual counts of squares in each step, assume that by counting the squares in the rectangles at step 0, step 1, step 2, we find that the number of squares in step 0 is \(a_0\), in step 1 is \(a_1\), and in step 2 is \(a_2\). If \(a_1 - a_0=a_2 - a_1 = d\), the rectangle grows by adding \(d\) squares in each step. The equation for the number of squares \(S(n)\) in step \(n\) is \(S(n)=a_0+nd\).
b. Let's say we find the number of squares in step 0 is \(a_0\), in step 1 is \(a_1\), and in step 2 is \(a_2\). If \(\frac{a_1}{a_0}=\frac{a_2}{a_1}=r\), the equation for the number of squares \(S(n)\) in step \(n\) is \(S(n)=a_0r^n\).
c. If the differences between consecutive terms are not constant (not arithmetic) and the ratios between consecutive terms are not constant (not geometric), then it is neither. We can check this by calculating the differences \(S(n + 1)-S(n)\) for different \(n\) values and the ratios \(\frac{S(n + 1)}{S(n)}\) for different \(n\) values. If the differences vary and the ratios vary, it is neither an arithmetic nor a geometric sequence.