I have been working on this problem for two days... I can only get as the characteristic part of the recurrence, I just can't figure out a proper guess for the particular solution.
$$T(n+2)-4T(n+1)+4T(n)=2^n+7$$ Where the characteristic part is $$T(n) = a\cdot2^n + b\cdot2\cdot n\cdot2^n$$
Where a and b are constants
$$a_{n+2}+4a_{n+1}-4a_n=2^n+7$$
$r-2=0$, $r=2$
$$a_n^{(\bar h)}=c_12^n+c_2n2^n$$
Guess: $\alpha2^n+\beta$
$$(\alpha2^n+\beta)+4(\alpha2^{n-1}+\beta)-4(\alpha2^{n-2}+\beta)$$
$$(\alpha2^n+\beta)+\alpha2^{n+1}+\underline{4\beta}-4\alpha2^{n-2}-\underline{4\beta}=2^n+7$$
$$\begin{align*} \alpha2^n+\alpha2^{n+1}-4\alpha2^{n-2}&=2^n\\ \beta&=7 \end{align*}$$
$$\underline{\alpha}+2\alpha-\underline{\alpha}=1$$
$$2\alpha=1$$
$$\alpha=\frac12$$
$$\begin{align*} a_n&=c_12^n+c_2n2^n+\frac122^n+7\\ &=c_12^n+c_2n2^n+2^{n-1}+7 \end{align*}$$
I have tried this, can anyone please tell me if I have committed a mistake and give me sugguestions on how to fix it.