Step 1

The given recurrence relation is

\(\displaystyle{x}_{{{n}}}-{3}{x}_{{{n}-{1}}}+{2}{x}_{{{n}-{2}}}={0}\)

The characteristic equation is

\(\displaystyle{r}^{{{2}}}-{3}{r}+{2}={0}\Rightarrow{\left({r}-{1}\right)}{\left({r}-{2}\right)}={0}\)

has the roots \(\displaystyle{r}_{{{1}}}={1}\) and \(\displaystyle{r}_{{{2}}}={2}\)

Step 2

Since the roots are distinct, the general solution is

\(\displaystyle{x}_{{{n}}}={c}_{{{1}}}{{r}_{{{1}}}^{{{n}}}}+{c}_{{{2}}}{{r}_{{{2}}}^{{{n}}}}\)

\(\displaystyle={c}_{{{1}}}{1}^{{{n}}}+{c}_{{{2}}}{2}^{{{n}}}\)

\(\displaystyle={c}_{{{1}}}+{c}_{{{2}}}{2}^{{{n}}}\)

Step 3

Plug the initial conditions to find the value of constants.

1) \(\displaystyle-{1}={x}_{{{1}}}={c}_{{{1}}}+{2}{c}_{{{2}}}\)

Also

2) \(\displaystyle{1}={x}_{{{2}}}={c}_{{{1}}}+{4}{c}_{{{2}}}\)

Step 4

Solving (1) and (2).

Subtract (1) from (2).

\(\displaystyle{2}{c}_{{{2}}}={2}\Rightarrow{c}_{{{2}}}={1}\)

From (1),

\(\displaystyle{c}_{{{1}}}=-{3}\)

Step 5

Thus the general solution becomes

\(\displaystyle{x}_{{{n}}}=-{3}+{2}^{{{n}}}\)

which is the general term.

The given recurrence relation is

\(\displaystyle{x}_{{{n}}}-{3}{x}_{{{n}-{1}}}+{2}{x}_{{{n}-{2}}}={0}\)

The characteristic equation is

\(\displaystyle{r}^{{{2}}}-{3}{r}+{2}={0}\Rightarrow{\left({r}-{1}\right)}{\left({r}-{2}\right)}={0}\)

has the roots \(\displaystyle{r}_{{{1}}}={1}\) and \(\displaystyle{r}_{{{2}}}={2}\)

Step 2

Since the roots are distinct, the general solution is

\(\displaystyle{x}_{{{n}}}={c}_{{{1}}}{{r}_{{{1}}}^{{{n}}}}+{c}_{{{2}}}{{r}_{{{2}}}^{{{n}}}}\)

\(\displaystyle={c}_{{{1}}}{1}^{{{n}}}+{c}_{{{2}}}{2}^{{{n}}}\)

\(\displaystyle={c}_{{{1}}}+{c}_{{{2}}}{2}^{{{n}}}\)

Step 3

Plug the initial conditions to find the value of constants.

1) \(\displaystyle-{1}={x}_{{{1}}}={c}_{{{1}}}+{2}{c}_{{{2}}}\)

Also

2) \(\displaystyle{1}={x}_{{{2}}}={c}_{{{1}}}+{4}{c}_{{{2}}}\)

Step 4

Solving (1) and (2).

Subtract (1) from (2).

\(\displaystyle{2}{c}_{{{2}}}={2}\Rightarrow{c}_{{{2}}}={1}\)

From (1),

\(\displaystyle{c}_{{{1}}}=-{3}\)

Step 5

Thus the general solution becomes

\(\displaystyle{x}_{{{n}}}=-{3}+{2}^{{{n}}}\)

which is the general term.