To determine which equation represents the sequence of diameters given by [tex]\(2.5 \, \text{cm}, 3.1 \, \text{cm}, 3.7 \, \text{cm}, 4.3 \, \text{cm}\)[/tex], we can follow a step-by-step analysis.
### Step 1: Identify the Pattern
First, let's observe the changes between consecutive terms to identify any pattern:
1. Between 2.5 and 3.1:
[tex]\(3.1 - 2.5 = 0.6\)[/tex]
2. Between 3.1 and 3.7:
[tex]\(3.7 - 3.1 = 0.6\)[/tex]
3. Between 3.7 and 4.3:
[tex]\(4.3 - 3.7 = 0.6\)[/tex]
We can see that each term increases by a constant difference of [tex]\(0.6\)[/tex]. This suggests that the sequence of diameters is an arithmetic sequence.
### Step 2: Determine the Initial Term and Common Difference
In an arithmetic sequence, the terms are given by:
[tex]\[ a, a+d, a+2d, a+3d, \ldots \][/tex]
Here:
- The initial term ([tex]\(a\)[/tex]) is [tex]\(2.5\)[/tex] cm.
- The common difference ([tex]\(d\)[/tex]) is [tex]\(0.6\)[/tex] cm.
### Step 3: Write the Equation for the nth Term
The general formula for the nth term of an arithmetic sequence is given by:
[tex]\[ f(n) = a + (n-1)d \][/tex]
Substituting the values of [tex]\(a\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ f(n) = 2.5 + (n-1) \times 0.6 \][/tex]
Simplifying this:
[tex]\[ f(n) = 2.5 + 0.6n - 0.6 \][/tex]
[tex]\[ f(n) = 0.6n + 1.9 \][/tex]
### Step 4: Compare with Given Options
The equation that represents our sequence is:
[tex]\[ f(n) = 0.6n + 1.9 \][/tex]
This matches the first option in the provided choices:
[tex]\[
\begin{array}{l}
f(n) = 0.6n + 1.9 \\
f(n) = 0.6n + 2.5 \\
f(n+1) = f(n) + 1.9 \\
f(n+1) = f(n) - 0.6 \\
\end{array}
\][/tex]
### Conclusion
Therefore, the equation that represents the sequence of diameters is:
[tex]\[ \boxed{f(n) = 0.6n + 1.9} \][/tex]
