1. **State the problem:** We are given a quadratic sequence: 8, 16, 26, 38, 52, ... and need to find the formula for the $n^{th}$ term. 2. **Recall the formula for a quadratic sequence:** The $n^{th}$ term of a quadratic sequence is generally given by $$a_n = An^2 + Bn + C$$ where $A$, $B$, and $C$ are constants to be determined. 3. **Find the first differences:** $$16 - 8 = 8$$ $$26 - 16 = 10$$ $$38 - 26 = 12$$ $$52 - 38 = 14$$ 4. **Find the second differences:** $$10 - 8 = 2$$ $$12 - 10 = 2$$ $$14 - 12 = 2$$ Since the second difference is constant and equals 2, this confirms the sequence is quadratic. 5. **Use the second difference to find $A$:** The second difference equals $2A$, so $$2A = 2 \implies A = 1$$ 6. **Write the general term with $A=1$:** $$a_n = n^2 + Bn + C$$ 7. **Use the first two terms to find $B$ and $C$:** For $n=1$, $a_1 = 8$: $$1^2 + B(1) + C = 8 \implies 1 + B + C = 8 \implies B + C = 7$$ For $n=2$, $a_2 = 16$: $$2^2 + B(2) + C = 16 \implies 4 + 2B + C = 16 \implies 2B + C = 12$$ 8. **Solve the system:** Subtract the first equation from the second: $$ (2B + C) - (B + C) = 12 - 7 \implies B = 5$$ Substitute $B=5$ into $B + C = 7$: $$5 + C = 7 \implies C = 2$$ 9. **Final formula:** $$a_n = n^2 + 5n + 2$$ This is the $n^{th}$ term rule for the sequence.