f(1) = 15,\quad f(2) = 30,\quad f(3) = 45,\quad f(4) = 60 - AIKO, infinite ways to autonomy.
Understanding the Linear Pattern: Solving for f(n) = 15 at f(1) = 15, f(2) = 30, f(3) = 45, f(4) = 60
Understanding the Linear Pattern: Solving for f(n) = 15 at f(1) = 15, f(2) = 30, f(3) = 45, f(4) = 60
When examining a sequence of ordered pairs such as:
- f(1) = 15
- f(2) = 30
- f(3) = 45
- f(4) = 60
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)
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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
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✓ 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
This predictable growth makes linear functions ideal for modeling steady, consistent change.
Step 4: Alternative Representations
Though f(n) = 15n is the simplest form, the same sequence can also be expressed using recursive definitions:
- Recursive form:
f(1) = 15
f(n) = f(n−1) + 15 for n > 1
This recursive pattern mirrors the additive growth visible in the table.
