f(1) = 15,\quad f(2) = 30,\quad f(3) = 45,\quad f(4) = 60 - go-checkin.com

Understanding the Context

we notice a clear arithmetic progression: each term increases by 15 as the input increases by 1. This strong linear pattern suggests that the function f(n) is linear in nature, describable by a simple equation of the form:
f(n) = an + b, where a is the slope and b is the y-intercept.

Step 1: Confirming the Linear Relationship

Let’s plug in values to find a and b.

Using f(1) = 15:
 a(1) + b = 15 → a + b = 15  (1)

Key Insights

Using f(2) = 30:
 a(2) + b = 30 → 2a + b = 30  (2)

Subtract (1) from (2):
 (2a + b) − (a + b) = 30 − 15
 a = 15

Substitute a = 15 into (1):
 15 + b = 15 → b = 0

Thus, the function is:
f(n) = 15n

Check with all points:

• f(1) = 15×1 = 15
  
• f(2) = 15×2 = 30
  
• f(3) = 15×3 = 45
  
• f(4) = 15×4 = 60

Final Thoughts

✓ Confirmed—this linear model fits perfectly.

Step 2: Real-World Meaning Behind the Pattern

Functions like f(n) = 15n model proportional growth, where the output increases steadily with the input. In practical terms, if f(n) represents a total value accumulating per time unit, then:

• After 1 unit: $15
  
• After 2 units: $30
  
• After 3 units: $45
  
• After 4 units: $60

This could represent an increasing bonus per hour, escalating rewards, or cumulative payments growing linearly with time.

Step 3: Predicting Future Values

Using the formula f(n) = 15n, you can easily compute future outputs:

• f(5) = 15×5 = $75
  
• f(6) = 15×6 = $90