1. **Problem statement:** We have a row of $n$ cubes, each cube is $1 \text{ cm}$ on each side. We want to find: a) An expression for the surface area of the shape made by joining $n$ cubes in a row. b) The number of cubes $n$ that gives a surface area of $254 \text{ cm}^2$. 2. **Understanding the shape and surface area:** Each cube has 6 faces, each face is $1 \times 1 = 1 \text{ cm}^2$. For $n$ cubes alone, total surface area if separate would be $6n$. 3. **Adjusting for joined faces:** When cubes are joined in a row, each pair of adjacent cubes shares a face, so those faces are not visible. Each shared face removes 2 faces from the total visible surface area (one from each cube). 4. **Formula for surface area:** - Total faces without joining: $6n$ - Number of shared faces: $n-1$ - Each shared face removes $2$ faces from visible surface area So, surface area $S$ is: $$ S = 6n - 2(n-1) = 6n - 2n + 2 = 4n + 2 $$ 5. **Check with example $n=7$:** $$ S = 4(7) + 2 = 28 + 2 = 30 \text{ cm}^2 $$ 6. **Find $n$ for $S=254$:** $$ 254 = 4n + 2 $$ Subtract 2: $$ 252 = 4n $$ Divide by 4: $$ n = \frac{252}{4} = 63 $$ **Final answers:** - a) Surface area expression: $S = 4n + 2$ - b) Number of cubes for $S=254$ is $n=63$