Question

1. write an expression that represents the total surface area (including the top and bottom) of a tower of n cubes each having a side length of s. (do not include faces that cover each other.)

Explanation:

Step1: Analyze n=1 case

For \( n = 1 \), the cube has 6 faces. Each face area is \( s^2 \), so surface area \( SA_1=6s^2 \).

Step2: Analyze n=2 case

For \( n = 2 \), two cubes are stacked. The total faces without overlapping: each cube has 6 faces, but 2 faces (one from each cube) are covered. So \( SA_2 = 2\times6s^2-2s^2=10s^2 \).

Step3: Analyze n=3 case

For \( n = 3 \), three cubes are stacked. Total faces without overlapping: \( 3\times6s^2 - 4s^2=14s^2 \).

Step4: Find the pattern

The sequence for surface area: when \( n = 1 \), 6; \( n = 2 \), 10; \( n = 3 \), 14. The difference between terms is 4. So the formula is \( SA = 4ns^2+2s^2=2s^2(2n + 1) \) or \( SA=(4n + 2)s^2 \).

Answer:

\(\boldsymbol{(4n + 2)s^2}\) (or equivalent forms like \(2s^2(2n + 1)\))