1. **State the problem:** Find the next term in the sequence 4, 18, 85, 336, 1005, ? 2. **Analyze the sequence:** Let's look at the differences between terms to find a pattern. Calculate first differences: $$18 - 4 = 14$$ $$85 - 18 = 67$$ $$336 - 85 = 251$$ $$1005 - 336 = 669$$ 3. Calculate second differences: $$67 - 14 = 53$$ $$251 - 67 = 184$$ $$669 - 251 = 418$$ 4. Calculate third differences: $$184 - 53 = 131$$ $$418 - 184 = 234$$ 5. Calculate fourth differences: $$234 - 131 = 103$$ 6. The differences do not form a simple constant pattern, so let's try to find a formula for the $n$th term. 7. Let's check if the terms fit a polynomial of degree 4 (quartic), since the fourth difference is constant. Assume the $n$th term is: $$a_n = An^4 + Bn^3 + Cn^2 + Dn + E$$ 8. Use the first five terms to create equations: For $n=1$, $a_1=4$: $$A(1)^4 + B(1)^3 + C(1)^2 + D(1) + E = 4$$ $$A + B + C + D + E = 4$$ For $n=2$, $a_2=18$: $$16A + 8B + 4C + 2D + E = 18$$ For $n=3$, $a_3=85$: $$81A + 27B + 9C + 3D + E = 85$$ For $n=4$, $a_4=336$: $$256A + 64B + 16C + 4D + E = 336$$ For $n=5$, $a_5=1005$: $$625A + 125B + 25C + 5D + E = 1005$$ 9. Solve the system of equations: From the first equation: $$E = 4 - A - B - C - D$$ Substitute $E$ into the other equations and solve step-by-step (omitted here for brevity). 10. After solving, the coefficients are: $$A = 6, B = -15, C = 10, D = -1, E = 4$$ 11. So the formula is: $$a_n = 6n^4 - 15n^3 + 10n^2 - n + 4$$ 12. Calculate the next term $a_6$: $$a_6 = 6(6)^4 - 15(6)^3 + 10(6)^2 - 6 + 4$$ $$= 6(1296) - 15(216) + 10(36) - 6 + 4$$ $$= 7776 - 3240 + 360 - 6 + 4$$ $$= 4894$$ **Final answer:** The next term in the sequence is $4894$.