Did I solve this linear homogeneous recurrence relation correctly?

Question

I'm not sure how to enter math on this site because I'm pretty new, but I typed my solution up on word.

My solutions:

Answer 1

You are writing down the wrong quadratic each time. For $a_n=7a_{n-1}-10a_{n-2}$ you want $t^2-7t+10=0$ so $a_n=B5^n+D2^n$.

Similarly for the other one the quadratic is $t^2-6t+9=0$. The second one has a repeated root. For that you need $a_n=B3^n+Dn3^n$.

Having got your values for $B,D$ I recommend that you check that they do indeed give the correct values for $a_0,a_1$ and that they look as though they will give integer values for all $a_n$.

[Added later]

Just to finish off the 2nd one: we have $1=a_0=B,1=a_1=3B+3D$, so $B=1,D=-\frac{2}{3}$ and hence $a_n=-(2n-3)3^{n-1}$.