1. Let's start by understanding the problem: you want to learn how to solve quadratic sequences, cubic sequences, and geometric sequences. 2. **Quadratic sequences** follow the pattern where the second differences are constant. The general term is given by $$a_n = An^2 + Bn + C$$ where $A$, $B$, and $C$ are constants. 3. To find $A$, $B$, and $C$, use the first few terms of the sequence and set up equations: - Calculate the first differences (subtract consecutive terms). - Calculate the second differences (subtract consecutive first differences). - The second difference equals $2A$. 4. **Example for quadratic:** If the second difference is constant and equals $d$, then $$2A = d \Rightarrow A = \frac{d}{2}$$. 5. Use the first term $a_1$ and the formula $$a_1 = A(1)^2 + B(1) + C = A + B + C$$ to find $B + C$. 6. Use the second term $a_2$ similarly: $$a_2 = 4A + 2B + C$$. 7. Solve the system of equations to find $B$ and $C$. 8. **Cubic sequences** have constant third differences. The general term is $$a_n = An^3 + Bn^2 + Cn + D$$. 9. Find the third difference, which equals $6A$, so $$A = \frac{\text{third difference}}{6}$$. 10. Use the first four terms to set up equations and solve for $B$, $C$, and $D$. 11. **Geometric sequences** have a constant ratio $r$ between terms. The general term is $$a_n = ar^{n-1}$$ where $a$ is the first term. 12. To find $r$, divide the second term by the first term: $$r = \frac{a_2}{a_1}$$. 13. Use the formula to find any term or sum of terms. This covers the basics of solving quadratic, cubic, and geometric sequences.