Let's determine the volume of the pyramid step-by-step given the details.
### Step 1: Identify the base area
The base of the pyramid is a square with edge length [tex]\( n \)[/tex] units. The formula for the area of a square is:
[tex]\[ \text{Base Area} = \text{side}^2 = n^2 \][/tex]
### Step 2: Identify the height
The height of the pyramid from the base to the apex is given as [tex]\( n - 1 \)[/tex] units.
### Step 3: Use the formula for the volume of a pyramid
The volume [tex]\( V \)[/tex] of a pyramid with a base area [tex]\( B \)[/tex] and height [tex]\( h \)[/tex] is given by:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
### Step 4: Substitute the base area and height into the volume formula
From the problem:
- Base Area [tex]\( B = n^2 \)[/tex]
- Height [tex]\( h = n - 1 \)[/tex]
Substituting these into the volume formula, we get:
[tex]\[ V = \frac{1}{3} \times n^2 \times (n - 1) \][/tex]
[tex]\[ V = \frac{1}{3} n^2 (n - 1) \][/tex]
### Step 5: Compare with the given options
Let's compare the expression we derived with the given options:
1. [tex]\( \frac{1}{3} n(n-1) \)[/tex] units[tex]\(^3\)[/tex]
2. [tex]\( \frac{1}{3} n(n-1)^2 \)[/tex] units[tex]\(^3\)[/tex]
3. [tex]\( \frac{1}{3} n^2(n-1) \)[/tex] units[tex]\(^3\)[/tex]
4. [tex]\( \frac{1}{3} n^3(n-1) \)[/tex] units[tex]\(^3\)[/tex]
The expression that matches our derived formula is:
[tex]\[ \frac{1}{3} n^2 (n - 1) \][/tex]
### Conclusion
Hence, the correct expression that represents the volume of the pyramid is:
[tex]\[
\boxed{\frac{1}{3} n^2(n-1)}
\][/tex] units[tex]\(^3\)[/tex].
