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

remolatg 2021-08-22 Answered
If \(\displaystyle{x}_{{{1}}}=-{1},\ {x}_{{{2}}}={1},\ {X}_{{{n}}}={3}{X}_{{{\left({n}-{1}\right)}}}-{\mid}{2}{X}_{{{\left({n}-{2}\right)}}},\ \forall{n}\geq{3}.\) Find the general term \(\displaystyle{X}_{{{n}}}\)

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Expert Answer

l1koV
Answered 2021-08-23 Author has 2895 answers
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.
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