Question

An= 14an-1 - 59an-2+ 70an-3

r^3-14r^2+59r-70
r= 2, 5, 7

Ao= 9
A1=32
A2= 144

Answer 1

I don't resolve your problem, just continue.

Answer 2

since you have r =2, 5, 7 , the answer form will have the form of \[A_n= C_12^n+C_25^n +C_37^n\]you have to plug value of A to the form to calculate value of C's
\[A_0=9 --> C_1*2^0+C_2*5^0+C_3*7^0=9\\C_1+C_2+C_3=9(equation1)\]
\[A_1=32---> C_1*2^1+C_2*5^1+C_3*7^1=32\\2C_1+5C_2+7C_3=32(equation2)\]
\[A_3=144---> C_1*2^3+C_2*5^3+C_3*7^3=144\\8C_1+125C_2+343C_3=144(equation3)\]
from the equations 1,2,3 you solve for \[C_1,C_2,C_3\]
then plug them back to \[A_n= C_12^n+C_25^n+C_37^n\]
That's the final answer.
Good luck. generating function needs time to be solved