Question

use the diagram below to answer questions 8a-8c.
diagrams: step 1 (two rectangles), step 2 (four rectangles), step 3 (eight?/six? rectangles)
8a. write the recursive equation for the box pattern, including the solution for f(1).
f(n) = _____________, f(1) = ___________
8b. write the explicit equation for the box pattern.
f(n) = _____________________
8c. is this an arithmetic or geometric sequence? explain your thinking?
blank lines for answer

Explanation:

8A

Step1: Count boxes in each step

Step 1 (n=1): 2 boxes. Step 2 (n=2): 4 boxes. Step 3 (n=3): 6 boxes.

Step2: Find recursive relation

The difference between consecutive terms: \( 4 - 2 = 2 \), \( 6 - 4 = 2 \). So \( f(n) = f(n - 1) + 2 \) for \( n \geq 2 \), and \( f(1) = 2 \).

Step1: Identify sequence type

From 8A, it’s an arithmetic sequence with first term \( a_1 = 2 \), common difference \( d = 2 \).

Step2: Use arithmetic explicit formula

The explicit formula for an arithmetic sequence is \( f(n) = a_1 + (n - 1)d \). Substituting \( a_1 = 2 \), \( d = 2 \):
\( f(n) = 2 + (n - 1) \times 2 = 2n \).

An arithmetic sequence has a constant common difference between consecutive terms. Here, \( f(1)=2 \), \( f(2)=4 \), \( f(3)=6 \). The difference \( f(2)-f(1)=2 \), \( f(3)-f(2)=2 \) (constant). A geometric sequence has a constant ratio (not the case here, \( 4/2 = 2 \), \( 6/4 = 1.5 \), ratio not constant). So it’s arithmetic.

Answer:

\( f(n) = f(n - 1) + 2 \) (for \( n \geq 2 \)), \( f(1) = 2 \)

8B