# If x_{1}=-1,\ x_{2}=1,\ X_{n}=3X_{(n-1)}-|2X_{(n-2)},\ \forall n\geq3. Find the gene

If Find the general term ${X}_{n}$
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Step 1
The given recurrence relation is
${x}_{n}-3{x}_{n-1}+2{x}_{n-2}=0$
The characteristic equation is
${r}^{2}-3r+2=0⇒\left(r-1\right)\left(r-2\right)=0$
has the roots ${r}_{1}=1$ and ${r}_{2}=2$
Step 2
Since the roots are distinct, the general solution is
${x}_{n}={c}_{1}{r}_{1}^{n}+{c}_{2}{r}_{2}^{n}$
$={c}_{1}{1}^{n}+{c}_{2}{2}^{n}$
$={c}_{1}+{c}_{2}{2}^{n}$
Step 3
Plug the initial conditions to find the value of constants.
1) $-1={x}_{1}={c}_{1}+2{c}_{2}$
Also
2) $1={x}_{2}={c}_{1}+4{c}_{2}$
Step 4
Solving (1) and (2).
Subtract (1) from (2).
$2{c}_{2}=2⇒{c}_{2}=1$
From (1),
${c}_{1}=-3$
Step 5
Thus the general solution becomes
${x}_{n}=-3+{2}^{n}$
which is the general term.