f(1) = 15,\quad f(2) = 30,\quad f(3) = 45,\quad 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)

$f(1) = 15,\quad f(2) = 30,\quad f(3) = 45,\quad f(4) = 60$

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

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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

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.