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Closed form for $t_n = 6t_n-1-9t_n-2$ where $t_0 = 5$ and $t_1 = 9$
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Consider the sequence defined by
$$
begincases
t_0=5\
t_1=9\
t_n=6t_n-1-9t_n-2 & textif nge 2
endcases
.$$
Find a closed form for $t_n$.
Your response should be a formula in terms of $n$, and should not contain terms such as $t_n,$ $t_n-1,$ and so on. Do not include $``t_n=text''$ in your response.
I tried forming a sequence by taking some values for $n$ and finding $t_n$. Once that was done, I moved on to find some pattern between $n$ and $t_n$ but couldn't find any. Here's the sequence:
$$beginarray
n & 0 & 1 & 2 & 3 & 4 \
hline
t_n & 5 & 9 & 9 & -27 & -243
endarray$$
What am I suppose to do?
sequences-and-series recurrence-relations
edited Sep 2 at 11:10
user21820
36.1k440140
asked Sep 2 at 8:31
WolverineA03
1427
add a comment |Â
up vote
1
down vote
favorite
Consider the sequence defined by
$$
begincases
t_0=5\
t_1=9\
t_n=6t_n-1-9t_n-2 & textif nge 2
endcases
.$$
Find a closed form for $t_n$.
Your response should be a formula in terms of $n$, and should not contain terms such as $t_n,$ $t_n-1,$ and so on. Do not include $``t_n=text''$ in your response.
I tried forming a sequence by taking some values for $n$ and finding $t_n$. Once that was done, I moved on to find some pattern between $n$ and $t_n$ but couldn't find any. Here's the sequence:
$$beginarray
n & 0 & 1 & 2 & 3 & 4 \
hline
t_n & 5 & 9 & 9 & -27 & -243
endarray$$
What am I suppose to do?
sequences-and-series recurrence-relations
edited Sep 2 at 11:10
user21820
36.1k440140
asked Sep 2 at 8:31
WolverineA03
1427
If you like, see Difference equation.
– xbh
Sep 2 at 8:34
4
Study the course material?
– amd
Sep 2 at 8:34
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider the sequence defined by
$$
begincases
t_0=5\
t_1=9\
t_n=6t_n-1-9t_n-2 & textif nge 2
endcases
.$$
Find a closed form for $t_n$.
Your response should be a formula in terms of $n$, and should not contain terms such as $t_n,$ $t_n-1,$ and so on. Do not include $``t_n=text''$ in your response.
I tried forming a sequence by taking some values for $n$ and finding $t_n$. Once that was done, I moved on to find some pattern between $n$ and $t_n$ but couldn't find any. Here's the sequence:
$$beginarray
n & 0 & 1 & 2 & 3 & 4 \
hline
t_n & 5 & 9 & 9 & -27 & -243
endarray$$
What am I suppose to do?
sequences-and-series recurrence-relations
edited Sep 2 at 11:10
user21820
36.1k440140
asked Sep 2 at 8:31
WolverineA03
1427
Consider the sequence defined by
$$
begincases
t_0=5\
t_1=9\
t_n=6t_n-1-9t_n-2 & textif nge 2
endcases
.$$
Find a closed form for $t_n$.
Your response should be a formula in terms of $n$, and should not contain terms such as $t_n,$ $t_n-1,$ and so on. Do not include $``t_n=text''$ in your response.
I tried forming a sequence by taking some values for $n$ and finding $t_n$. Once that was done, I moved on to find some pattern between $n$ and $t_n$ but couldn't find any. Here's the sequence:
$$beginarray
n & 0 & 1 & 2 & 3 & 4 \
hline
t_n & 5 & 9 & 9 & -27 & -243
endarray$$
What am I suppose to do?
sequences-and-series recurrence-relations
edited Sep 2 at 11:10
user21820
36.1k440140
asked Sep 2 at 8:31
WolverineA03
1427
edited Sep 2 at 11:10
user21820
36.1k440140
edited Sep 2 at 11:10
user21820
36.1k440140
edited Sep 2 at 11:10
user21820
36.1k440140
36.1k440140
asked Sep 2 at 8:31
WolverineA03
1427
asked Sep 2 at 8:31
WolverineA03
1427
asked Sep 2 at 8:31
WolverineA03
1427
1427
If you like, see Difference equation.
– xbh
Sep 2 at 8:34
4
Study the course material?
– amd
Sep 2 at 8:34
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44
add a comment |Â
If you like, see Difference equation.
– xbh
Sep 2 at 8:34
4
Study the course material?
– amd
Sep 2 at 8:34
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44
If you like, see Difference equation.
– xbh
Sep 2 at 8:34
If you like, see Difference equation.
– xbh
Sep 2 at 8:34
4
4
Study the course material?
– amd
Sep 2 at 8:34
Study the course material?
– amd
Sep 2 at 8:34
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
6
down vote
Note that the characteristic polynomial of your homogeneous linear recurrence is $z^2-6z+9=(z-3)^2$. Therefore the solution has the form
$$t_n=(An+B)3^n$$
where $A$ and $B$ are constants to be found.
answered Sep 2 at 8:41
Robert Z
85.5k1055123
add a comment |Â
up vote
3
down vote
Another widely used approach is using generating functions, i.e.
$$f(x)=sumlimits_n=0colorredt_nx^n=
5+9x+sumlimits_n=2t_nx^n=
5+9x+sumlimits_n=2left(6t_n-1-9t_n-2right)x^n=\
5+9x+6xleft(sumlimits_n=2t_n-1x^n-1right)-9x^2left(sumlimits_n=2t_n-2x^n-2right)=\
5+9x+6xleft(sumlimits_n=1t_nx^nright)-9x^2left(sumlimits_n=0t_nx^nright)=
5+9x+6xleft(f(x)-5right)-9x^2f(x)$$
or
$$f(x)=5+9x+6xleft(f(x)-5right)-9x^2f(x) iff \
f(x)=frac5-21x1-6x+9x^2=
frac5-21x(1-3x)^2=
frac71-3x-frac2(1-3x)^2=\
7left(sumlimits_n=03^nx^nright)-2left(sumlimits_n=0(n+1)3^nx^nright)=\
sumlimits_n=0colorredleft(7-2(n+1)right)3^nx^n$$
and
$$t_n=left(5-2nright)3^n$$
answered Sep 2 at 10:27
rtybase
9,10721433
add a comment |Â
up vote
2
down vote
Hint. Make the ansatz $$t_n=q^n$$
and for your work: the solution is given by
$$t_n=3^n(5-2n)$$
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
6
down vote
Note that the characteristic polynomial of your homogeneous linear recurrence is $z^2-6z+9=(z-3)^2$. Therefore the solution has the form
$$t_n=(An+B)3^n$$
where $A$ and $B$ are constants to be found.
answered Sep 2 at 8:41
Robert Z
85.5k1055123
add a comment |Â
up vote
6
down vote
Note that the characteristic polynomial of your homogeneous linear recurrence is $z^2-6z+9=(z-3)^2$. Therefore the solution has the form
$$t_n=(An+B)3^n$$
where $A$ and $B$ are constants to be found.
answered Sep 2 at 8:41
Robert Z
85.5k1055123
add a comment |Â
up vote
6
down vote
up vote
6
down vote
Note that the characteristic polynomial of your homogeneous linear recurrence is $z^2-6z+9=(z-3)^2$. Therefore the solution has the form
$$t_n=(An+B)3^n$$
where $A$ and $B$ are constants to be found.
answered Sep 2 at 8:41
Robert Z
85.5k1055123
Note that the characteristic polynomial of your homogeneous linear recurrence is $z^2-6z+9=(z-3)^2$. Therefore the solution has the form
$$t_n=(An+B)3^n$$
where $A$ and $B$ are constants to be found.
answered Sep 2 at 8:41
Robert Z
85.5k1055123
answered Sep 2 at 8:41
Robert Z
85.5k1055123
answered Sep 2 at 8:41
Robert Z
85.5k1055123
answered Sep 2 at 8:41
Robert Z
85.5k1055123
85.5k1055123
add a comment |Â
add a comment |Â
up vote
3
down vote
Another widely used approach is using generating functions, i.e.
$$f(x)=sumlimits_n=0colorredt_nx^n=
5+9x+sumlimits_n=2t_nx^n=
5+9x+sumlimits_n=2left(6t_n-1-9t_n-2right)x^n=\
5+9x+6xleft(sumlimits_n=2t_n-1x^n-1right)-9x^2left(sumlimits_n=2t_n-2x^n-2right)=\
5+9x+6xleft(sumlimits_n=1t_nx^nright)-9x^2left(sumlimits_n=0t_nx^nright)=
5+9x+6xleft(f(x)-5right)-9x^2f(x)$$
or
$$f(x)=5+9x+6xleft(f(x)-5right)-9x^2f(x) iff \
f(x)=frac5-21x1-6x+9x^2=
frac5-21x(1-3x)^2=
frac71-3x-frac2(1-3x)^2=\
7left(sumlimits_n=03^nx^nright)-2left(sumlimits_n=0(n+1)3^nx^nright)=\
sumlimits_n=0colorredleft(7-2(n+1)right)3^nx^n$$
and
$$t_n=left(5-2nright)3^n$$
answered Sep 2 at 10:27
rtybase
9,10721433
add a comment |Â
up vote
3
down vote
Another widely used approach is using generating functions, i.e.
$$f(x)=sumlimits_n=0colorredt_nx^n=
5+9x+sumlimits_n=2t_nx^n=
5+9x+sumlimits_n=2left(6t_n-1-9t_n-2right)x^n=\
5+9x+6xleft(sumlimits_n=2t_n-1x^n-1right)-9x^2left(sumlimits_n=2t_n-2x^n-2right)=\
5+9x+6xleft(sumlimits_n=1t_nx^nright)-9x^2left(sumlimits_n=0t_nx^nright)=
5+9x+6xleft(f(x)-5right)-9x^2f(x)$$
or
$$f(x)=5+9x+6xleft(f(x)-5right)-9x^2f(x) iff \
f(x)=frac5-21x1-6x+9x^2=
frac5-21x(1-3x)^2=
frac71-3x-frac2(1-3x)^2=\
7left(sumlimits_n=03^nx^nright)-2left(sumlimits_n=0(n+1)3^nx^nright)=\
sumlimits_n=0colorredleft(7-2(n+1)right)3^nx^n$$
and
$$t_n=left(5-2nright)3^n$$
answered Sep 2 at 10:27
rtybase
9,10721433
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Another widely used approach is using generating functions, i.e.
$$f(x)=sumlimits_n=0colorredt_nx^n=
5+9x+sumlimits_n=2t_nx^n=
5+9x+sumlimits_n=2left(6t_n-1-9t_n-2right)x^n=\
5+9x+6xleft(sumlimits_n=2t_n-1x^n-1right)-9x^2left(sumlimits_n=2t_n-2x^n-2right)=\
5+9x+6xleft(sumlimits_n=1t_nx^nright)-9x^2left(sumlimits_n=0t_nx^nright)=
5+9x+6xleft(f(x)-5right)-9x^2f(x)$$
or
$$f(x)=5+9x+6xleft(f(x)-5right)-9x^2f(x) iff \
f(x)=frac5-21x1-6x+9x^2=
frac5-21x(1-3x)^2=
frac71-3x-frac2(1-3x)^2=\
7left(sumlimits_n=03^nx^nright)-2left(sumlimits_n=0(n+1)3^nx^nright)=\
sumlimits_n=0colorredleft(7-2(n+1)right)3^nx^n$$
and
$$t_n=left(5-2nright)3^n$$
answered Sep 2 at 10:27
rtybase
9,10721433
Another widely used approach is using generating functions, i.e.
$$f(x)=sumlimits_n=0colorredt_nx^n=
5+9x+sumlimits_n=2t_nx^n=
5+9x+sumlimits_n=2left(6t_n-1-9t_n-2right)x^n=\
5+9x+6xleft(sumlimits_n=2t_n-1x^n-1right)-9x^2left(sumlimits_n=2t_n-2x^n-2right)=\
5+9x+6xleft(sumlimits_n=1t_nx^nright)-9x^2left(sumlimits_n=0t_nx^nright)=
5+9x+6xleft(f(x)-5right)-9x^2f(x)$$
or
$$f(x)=5+9x+6xleft(f(x)-5right)-9x^2f(x) iff \
f(x)=frac5-21x1-6x+9x^2=
frac5-21x(1-3x)^2=
frac71-3x-frac2(1-3x)^2=\
7left(sumlimits_n=03^nx^nright)-2left(sumlimits_n=0(n+1)3^nx^nright)=\
sumlimits_n=0colorredleft(7-2(n+1)right)3^nx^n$$
and
$$t_n=left(5-2nright)3^n$$
answered Sep 2 at 10:27
rtybase
9,10721433
answered Sep 2 at 10:27
rtybase
9,10721433
answered Sep 2 at 10:27
rtybase
9,10721433
answered Sep 2 at 10:27
rtybase
9,10721433
9,10721433
add a comment |Â
add a comment |Â
up vote
2
down vote
Hint. Make the ansatz $$t_n=q^n$$
and for your work: the solution is given by
$$t_n=3^n(5-2n)$$
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
add a comment |Â
up vote
2
down vote
Hint. Make the ansatz $$t_n=q^n$$
and for your work: the solution is given by
$$t_n=3^n(5-2n)$$
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Hint. Make the ansatz $$t_n=q^n$$
and for your work: the solution is given by
$$t_n=3^n(5-2n)$$
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
Hint. Make the ansatz $$t_n=q^n$$
and for your work: the solution is given by
$$t_n=3^n(5-2n)$$
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
edited Sep 2 at 8:51
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
answered Sep 2 at 8:36
Dr. Sonnhard Graubner
68.2k32760
68.2k32760
add a comment |Â
add a comment |Â
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If you like, see Difference equation.
– xbh
Sep 2 at 8:34
4
Study the course material?
– amd
Sep 2 at 8:34
More seriously, have you studied any methods of solving such recurrences besides looking at first-order differences and trying to find a pattern?
– amd
Sep 2 at 8:44