5,14,22,29, write the next 2 numbers in this pattern and the pattern rule and the 8th number
P(n)Δ1Δ2P(1)=59P(2)=14−18P(3)=22−17P(4)=29−16P(5)=35−15P(6)=40−14P(7)=44−13P(8)=47−1
pattern rule: P(n)=an2+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
pattern rule: P(n)=−0.5∗n2+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
