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Consider the nonhomogeneous linear recurrence relation $a_n$=$3a_{n-1}$+$2^n$

in the book solution is given $a_n$=$-2^{n+1}$

but I’m getting $a_n$=$3^{n+1}-2^{n+1}$

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what is the base condition ?
$-2^{n+1}$ is the particular solution..
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base condition is $a_0$=1
im getting the same

homogeneous part i got $3^{n+1}$
and particular part $-2^{n+1}$
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You can verify yourself by taking small values of 'n'.

For eg.,  for $n=1$, recurrence relation gives $a_1=3a_0+2^1=3*1+2=5$

Now,$-2^{n+1}$ will give answer as $-2^2=-4$ where as $3^{n+1}-2^{n+1}$ will give answer as $3^2-2^2=5$. So, your answer is correct.