1. **State the problem:** We have a pattern of L-shaped figures made of tiles for values $n=1, 2, 3$. We want to find the number of tiles when $n=4$ and express the number of tiles as a function of $n$. 2. **Analyze the given data:** - For $n=1$, number of tiles = 3 - For $n=2$, number of tiles = 8 - For $n=3$, number of tiles = 15 3. **Look for a pattern or formula:** Notice the number of tiles for each $n$: $$3, 8, 15$$ Calculate the differences: $$8 - 3 = 5$$ $$15 - 8 = 7$$ The differences increase by 2 each time, suggesting a quadratic pattern. 4. **Assume a quadratic formula:** $$T(n) = an^2 + bn + c$$ Use the known values to find $a$, $b$, and $c$: - For $n=1$: $$a(1)^2 + b(1) + c = 3$$ $$a + b + c = 3$$ - For $n=2$: $$4a + 2b + c = 8$$ - For $n=3$: $$9a + 3b + c = 15$$ 5. **Solve the system:** From the first equation: $$c = 3 - a - b$$ Substitute into the second: $$4a + 2b + 3 - a - b = 8$$ $$3a + b + 3 = 8$$ $$3a + b = 5$$ Substitute into the third: $$9a + 3b + 3 - a - b = 15$$ $$8a + 2b + 3 = 15$$ $$8a + 2b = 12$$ Divide by 2: $$4a + b = 6$$ 6. **Solve for $a$ and $b$:** From: $$3a + b = 5$$ $$4a + b = 6$$ Subtract first from second: $$(4a + b) - (3a + b) = 6 - 5$$ $$a = 1$$ Substitute $a=1$ into $3a + b = 5$: $$3(1) + b = 5$$ $$b = 2$$ 7. **Find $c$:** $$c = 3 - a - b = 3 - 1 - 2 = 0$$ 8. **Final formula:** $$T(n) = n^2 + 2n$$ 9. **Calculate for $n=4$:** $$T(4) = 4^2 + 2(4) = 16 + 8 = 24$$ **Answer:** The number of tiles when $n=4$ is 24. **Summary:** The number of tiles in the pattern for any $n$ is given by $$T(n) = n^2 + 2n$$. For $n=4$, the number of tiles is 24.