1. **State the problem:** We are given the sequence 10, 15, 25, 40, ... and need to find the generator formula of the form $An^2 + Bn + C = a_n$ where $a_n$ is the $n$th term. 2. **Write the terms with their indices:** $$a_1 = 10, \quad a_2 = 15, \quad a_3 = 25, \quad a_4 = 40$$ 3. **Set up equations using the formula:** $$A(1)^2 + B(1) + C = 10$$ $$A(2)^2 + B(2) + C = 15$$ $$A(3)^2 + B(3) + C = 25$$ 4. **Simplify the equations:** $$A + B + C = 10$$ $$4A + 2B + C = 15$$ $$9A + 3B + C = 25$$ 5. **Subtract the first equation from the second and third to eliminate $C$:** $$\cancel{4A} + 2B + \cancel{C} - (\cancel{A} + B + \cancel{C}) = 15 - 10$$ $$3A + B = 5$$ $$\cancel{9A} + 3B + \cancel{C} - (\cancel{A} + B + \cancel{C}) = 25 - 10$$ $$8A + 2B = 15$$ 6. **Simplify the second equation by dividing by 2:** $$4A + B = \frac{15}{2}$$ 7. **Now solve the system:** $$3A + B = 5$$ $$4A + B = 7.5$$ 8. **Subtract the first from the second:** $$(4A + B) - (3A + B) = 7.5 - 5$$ $$A = 2.5$$ 9. **Substitute $A=2.5$ into $3A + B = 5$:** $$3(2.5) + B = 5$$ $$7.5 + B = 5$$ $$B = 5 - 7.5 = -2.5$$ 10. **Substitute $A$ and $B$ into $A + B + C = 10$ to find $C$:** $$2.5 - 2.5 + C = 10$$ $$C = 10$$ 11. **Final formula:** $$a_n = 2.5n^2 - 2.5n + 10$$ 12. **Check with $n=4$:** $$a_4 = 2.5(4)^2 - 2.5(4) + 10 = 2.5(16) - 10 + 10 = 40$$ This matches the given term. **Answer:** $$\boxed{a_n = 2.5n^2 - 2.5n + 10}$$