5,14,22,29, write the next 2 numbers in this pattern and the pattern rule and the 8th number

5,14,22,29, write the next 2 numbers in this pattern and the pattern rule and the 8th number

$$\begin{array}{|l|c|c|} \hline P(n) &\Delta_1 &\Delta_2\\ \hline P(1) = 5 & & \\ &9& \\ P(2) = 14 & & -1 \\ &8& \\ P(3) = 22 & & -1 \\ &7& \\ P(4) = 29 & & -1 \\ &6& \\ P(5) = 35 & & -1 \\ &5& \\ P(6) = 40 & & -1 \\ &4& \\ P(7) = 44& & -1 \\ &3& \\ P(8) = 47& & -1 \\ \hline \end{array}$$

$$\small{\text{ pattern rule: }} P(n)=an^2+bn+c$$

$$\\\small{\text{ (1) n=1 : }} \quad a*1+b*1+c=5\\ \small{\text{ (2) n=2 : }} \quad a*4+b*2+c=14\\ \small{\text{ (3) n=3 : }} \quad a*9+b*3+c=22 \\ \hline \small{\text{ (2)-(1) : }} \quad (4a-a) + (2b-b) +(c-c) = 14-5\\ \small{\text{ (4) : }} \quad 3a + b = 9\\ \hline \small{\text{ (3)-(2) : }} \quad (9a-4a) + (3b-2b) +(c-c) = 22-14\\ \small{\text{ (5) : }} \quad 5a + b = 8\\ \hline \small{\text{ (5)-(4) : }} (5a-3a) + (b-b) = 8-9\\ \small{\text{ a : }} 2a = -1 \quad a= -0.5 \\ \small{\text{ b : }} 3a+b=9 \quad 3*(-0.5) +b = 9 \quad b=10.5 \\ \small{\text{ c : }} a+b+c=5 \quad -0.5+10.5+c=5 \quad c=-5$$

$$\small{\text{ pattern rule: }} P(n)=-0.5*n^2+10.5*n-5$$

P(5) = -0.5 * 25 + 10.5 * 5 - 5 = 35

P(6) = -0.5 * 36 + 10.5 * 6 - 5 = 40

P(8) = -0.5 * 64 + 10.5 * 8 - 5 = 47