Question

part 2 – explore: worker growth (geometric sequence)
the number of workers on the site also follows a pattern. as the project grows, more teams join each week:
week number of workers
1 5
2 10
3 20
4 40
4: how does the number of workers change from one week to the next? is the change by addition or multiplication?
5: predict how many workers will be on - site in week 5, 6, and week 20

1. write both a recursive and explicit rule to describe the pattern.
2. how might recognizing arithmetic and geometric patterns help engineers and project managers predict construction materials, workforce needs, or budgeting?

Explanation:

Question 5: Predict workers in Week 5, 6, 20

Step1: Identify the pattern

The number of workers follows a geometric sequence. From the table: Week 1: 5, Week 2: 10, Week 3: 20, Week 4: 40. The common ratio \( r = \frac{10}{5} = 2 \), \( \frac{20}{10} = 2 \), etc. So the formula for a geometric sequence is \( a_n = a_1 \times r^{n - 1} \), where \( a_1 = 5 \), \( r = 2 \).

Step2: Calculate Week 5

For Week \( n = 5 \):
\( a_5 = 5 \times 2^{5 - 1} = 5 \times 2^4 = 5 \times 16 = 80 \)? Wait, no—Wait, Week 4 is 40. Wait, maybe \( a_1 = 5 \) (Week 1), Week 2: \( 5 \times 2 = 10 \), Week 3: \( 10 \times 2 = 20 \), Week 4: \( 20 \times 2 = 40 \), so Week \( n \): \( a_n = 5 \times 2^{n - 1} \). Wait, but Week 4: \( 5 \times 2^{3} = 40 \), correct. Then Week 5: \( 5 \times 2^{4} = 5 \times 16 = 80 \)? But in the handwritten note, Week 5 is 80×2=160? Wait, maybe I misread. Wait the table: Week 1:5, Week2:10, Week3:20, Week4:40. Then Week 5: 40×2=80? Wait the handwritten says "80×2 160"—oh, maybe the initial term is different. Wait the user's handwritten: Week 1:5×2=10? No, Week 1:5, Week2:10 (5×2), Week3:20 (10×2), Week4:40 (20×2), Week5:40×2=80? But the handwritten has "80×2 160"—maybe a miscalculation. Wait, let's re-express:

Wait the table as per the image: Week 1:5, Week2:10, Week3:20, Week4:40. So the sequence is \( a_n = 5 \times 2^{n - 1} \). Wait Week 4: \( 5 \times 2^{3} = 40 \), correct. Then Week 5: \( 5 \times 2^{4} = 80 \). But the handwritten note has "80×2 160"—maybe the user made a mistake, or maybe the initial term is 5, but Week 1 is 5, Week 2 is 10 (5×2), Week 3:20 (10×2), Week4:40 (20×2), Week5:80 (40×2), Week6:160 (80×2). Wait the handwritten says "160×2 320" for Week 6. Wait, maybe the user's initial term is Week 1:5, but the handwritten calculations show Week 5:80×2=160—so maybe the sequence is \( a_n = 5 \times 2^{n} \)? Wait Week 1:5×2^1=10? No, that doesn't match. Wait the table in the image: Week 1:5, Week2:10, Week3:20, Week4:40. So the correct formula is \( a_n = 5 \times 2^{n - 1} \). So:

• Week 5: \( n = 5 \), \( a_5 = 5 \times 2^{4} = 80 \)
• Week 6: \( n = 6 \), \( a_6 = 5 \times 2^{5} = 160 \)
• Week 20: \( n = 20 \), \( a_{20} = 5 \times 2^{19} \). Let's calculate \( 2^{10} = 1024 \), \( 2^{19} = 2^{10} \times 2^9 = 1024 \times 512 = 524,288 \). Then \( 5 \times 524,288 = 2,621,440 \).

Wait but the handwritten note has Week 5:160, Week 6:320—maybe the user considered Week 1 as 5, but the sequence is \( a_n = 5 \times 2^{n} \)? No, Week 1:5×2^1=10 (but table says 5). So there's a discrepancy. Wait the table in the image: Week 1:5, Week2:10, Week3:20, Week4:40. So the common ratio is 2, first term \( a_1 = 5 \). So:

Week 5: \( 40 \times 2 = 80 \)
Week 6: \( 80 \times 2 = 160 \)
Week 20: \( a_{20} = 5 \times 2^{19} = 2,621,440 \)

Step3: Verify with the handwritten

The handwritten has "80×2 160" (Week 5: 80×2=160? No, Week 4 is 40, so Week 5 should be 40×2=80. Then Week 5:80, Week 6:80×2=160, Week 20: \( 5 \times 2^{19} = 2,621,440 \).

Step1: Recursive Rule

In a geometric sequence, the recursive rule is \( a_n = a_{n - 1} \times r \), with initial term \( a_1 \). Here, \( a_1 = 5 \), \( r = 2 \). So:
Recursive: \( a_1 = 5 \), \( a_n = a_{n - 1} \times 2 \) for \( n > 1 \).

Step2: Explicit Rule

The explicit formula for a geometric sequence is \( a_n = a_1 \times r^{n - 1} \). Substituting \( a_1 = 5 \), \( r = 2 \):
\( a_n = 5 \times 2^{n - 1} \)

Recognizing arithmetic/geometric patterns helps in forecasting. For workforce: predict labor needs (e.g., if workers grow geometrically, plan staffing, equipment, and materials. For materials: if usage follows a pattern, order supplies in advance. Budgeting: project costs (labor, materials) over time to allocate funds, avoid shortages/overspending. Engineers use it for resource planning, project managers for scheduling and cost control.

Answer:

Week 5: 80 workers
Week 6: 160 workers
Week 20: \( 2,621,440 \) workers

Question 6: Recursive and Explicit Rules