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
