If ${x}_{1}=-1,\text{}{x}_{2}=1,\text{}{X}_{n}=3{X}_{(n-1)}-\mid 2{X}_{(n-2)},\text{}\mathrm{\forall}n\ge 3.$ Find the general term $X}_{n$

remolatg
2021-08-22
Answered

If ${x}_{1}=-1,\text{}{x}_{2}=1,\text{}{X}_{n}=3{X}_{(n-1)}-\mid 2{X}_{(n-2)},\text{}\mathrm{\forall}n\ge 3.$ Find the general term $X}_{n$

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l1koV

Answered 2021-08-23
Author has **100** answers

Step 1

The given recurrence relation is

${x}_{n}-3{x}_{n-1}+2{x}_{n-2}=0$

The characteristic equation is

${r}^{2}-3r+2=0\Rightarrow (r-1)(r-2)=0$

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).

$2{c}_{2}=2\Rightarrow {c}_{2}=1$

From (1),

${c}_{1}=-3$

Step 5

Thus the general solution becomes

$x}_{n}=-3+{2}^{n$

which is the general term.

The given recurrence relation is

The characteristic equation is

has the roots

Step 2

Since the roots are distinct, the general solution is

Step 3

Plug the initial conditions to find the value of constants.

1)

Also

2)

Step 4

Solving (1) and (2).

Subtract (1) from (2).

From (1),

Step 5

Thus the general solution becomes

which is the general term.

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