Subjects algebra

Green White Tiles 950604

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1. **Stating the problem:** We have a sequence of figures where green octagon tiles form an $n \times n$ block with white square gaps between them. 2. **Understanding the pattern:** - Figur 1 has 1 green tile. - Figur 2 has 4 green tiles arranged in a $2 \times 2$ block. - Figur 3 has 9 green tiles arranged in a $3 \times 3$ block. 3. **a) Number of green tiles in Figur 5:** Since the green tiles form an $n \times n$ block, the number of green tiles in Figur 5 is: $$ 5^2 = 25 $$ 4. **b) Number of green tiles in Figur $n$:** Generalizing, the number of green tiles in Figur $n$ is: $$ n^2 $$ 5. **c) Number of white squares in Figur 5:** The white squares appear between the green tiles, forming a grid of $(n-1) \times (n-1)$ white squares. For Figur 5: $$ (5-1)^2 = 4^2 = 16 $$ 6. **d) Number of white squares in Figur $n$:** Generalizing, the number of white squares in Figur $n$ is: $$ (n-1)^2 $$ **Summary:** - Green tiles in Figur 5: 25 - Green tiles in Figur $n$: $n^2$ - White squares in Figur 5: 16 - White squares in Figur $n$: $(n-1)^2$