![can someone tell me they got characteristic equation from the given recursive equation.][1] i know how to do the rest of problem but getting characteristic equation stopped me.
The recurrence is $a_n=3a_{n-1}+4a_{n-2}+5a_{n-3}$ The characteristic equation is $r^3-3r^2-4r-5=0$
This is a classic case of how to solve a linear recurrence relation. Note that any multiple of a solution or any sum of solutions will again be a solution, so we have a vector space of solutions. We guess that a solution is of the form $r^n$ and plug into the equation. This gives $r^3=3r^2+4r+5$, which is exactly the characteristic equation you give. It has three solutions, each of which gives a basis vector of the space of solutions to the original equation. The most general solution will be a sum of three terms, each a constant multiplied by $r^n$. Special rules apply if a root is repeated.
The cubic has one real and two imaginary roots. They are not rational. Alpha can find them, presumably using Cardano's method.