Question

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.)

Short answer

Total surface area of a tower of n cubes$\left(S_n\right)$ each having a side length of s is:

$S_n=4s^2\left(n\right)+2s^2$.

Step-by-step solution

Step 1. State the concept of total surface are of cube.

The total surface area of a cube$=6s^2$where s is the length of the side of each edge of the cube.

Step 2. Find the total surface area of the given figures.

Total surface area of the figure with 1 cube,

$$\begin{gathered} \left(S_1\right)=6\times s^2 \\ =6s^2 \end{gathered}$$.

Total surface area of the figure with 2 cubes.

$$\begin{gathered} \left(S_2\right)=\left(5\times s^2\right)+\left(5\times s^2\right) \\ =5s^2+5s^2 \\ =10s^2 \end{gathered}$$

Total surface area of the figure with 3 cubes is given as:

$$\begin{gathered} \left(S_3\right)=\left(5\times s^2\right)+\left(4\times s^2\right)+\left(5\times s^2\right) \\ =5s^2+4s^2+5s^2 \\ =14s^2 \end{gathered}$$

Therefore the total surface areas are$6s^2,10s^2\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}14s^2$ respectively.

Step 3. Identify the pattern in which the total surface area are changing with the increase of 1 cube in each figures.

First find the difference between the total surface areas of corresponding figures.

$$\begin{gathered} d=S_2-S_1 \\ =10s^2-6s^2 \\ =4s^2 \end{gathered}$$

And,

$$\begin{gathered} d=S_3-S_2 \\ =14s^2-10s^2 \\ =4s^2 \end{gathered}$$

Notice that the common difference is constant. Therefore the sequence of total surface area are in Arithmetic progression.

Step 4. Find the general formula for the total surface area of n cubes.

Total surface area of a tower of n cubes $\left(S_n\right)$each having a side length of s is given as follows.

$$\begin{gathered} S_n=S_1+\left(n-1\right)d \\ =6s^2+\left(n-1\right)4s^2 \\ =6s^2+4s^2\left(n\right)-4s^2 \\ {S}_n=4s^2\left(n\right)+2s^2 \end{gathered}$$

Hence the required expression.