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Answer:
Geometric Sequence
Step-by-step explanation:
1. Check the difference.
The difference between the 1st and 2nd term
The difference between the 2nd and 3rd term
The difference is not the same. Therefore, it is not an arithmetic sequence.
2. Check the ratio
The ratio between the 1st and 2nd term
The ratio between the 2nd and 3rd term
The ratio is the same. Therefore, it is a geometric sequence.
Take the homogeneous part and find the roots to the characteristic equation:
This means the characteristic solution is
.
Since the characteristic solution already contains both functions on the RHS of the ODE, you could try finding a solution via the method of undetermined coefficients of the form
. Finding the second derivative involves quite a few applications of the product rule, so I'll resort to a different method via variation of parameters.
With
and 
, you're looking for a particular solution of the form 
. The functions 
satisfy
where
is the Wronskian determinant of the two characteristic solutions.
So you have
So you end up with a solution
but since
is already accounted for in the characteristic solution, the particular solution is then
so that the general solution is
This means the characteristic solution is
Since the characteristic solution already contains both functions on the RHS of the ODE, you could try finding a solution via the method of undetermined coefficients of the form
With
where
So you have
So you end up with a solution
but since
so that the general solution is