Mixing
Finding sum of terms in a sequence
A sequence has $a_1=-2$ and $a_2=4$ and in the sequence, when $n>2$, $a_{n}$= $dfrac {a_{n-1}}{a_{n-2}}$. Find the sum of the first $99$ terms of sequence.
I tried to deduce from the patterns produced but couldn't draw any fruitful conclusions. $a_3=-2$, $a_4=-0.5$, $a_5=0.25$ and so on...
real-analysis sequences-and-series analysis arithmetic
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
add a comment |
A sequence has $a_1=-2$ and $a_2=4$ and in the sequence, when $n>2$, $a_{n}$= $dfrac {a_{n-1}}{a_{n-2}}$. Find the sum of the first $99$ terms of sequence.
I tried to deduce from the patterns produced but couldn't draw any fruitful conclusions. $a_3=-2$, $a_4=-0.5$, $a_5=0.25$ and so on...
real-analysis sequences-and-series analysis arithmetic
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
add a comment |
A sequence has $a_1=-2$ and $a_2=4$ and in the sequence, when $n>2$, $a_{n}$= $dfrac {a_{n-1}}{a_{n-2}}$. Find the sum of the first $99$ terms of sequence.
I tried to deduce from the patterns produced but couldn't draw any fruitful conclusions. $a_3=-2$, $a_4=-0.5$, $a_5=0.25$ and so on...
real-analysis sequences-and-series analysis arithmetic
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
A sequence has $a_1=-2$ and $a_2=4$ and in the sequence, when $n>2$, $a_{n}$= $dfrac {a_{n-1}}{a_{n-2}}$. Find the sum of the first $99$ terms of sequence.
I tried to deduce from the patterns produced but couldn't draw any fruitful conclusions. $a_3=-2$, $a_4=-0.5$, $a_5=0.25$ and so on...
real-analysis sequences-and-series analysis arithmetic
real-analysis sequences-and-series analysis arithmetic
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
edited Nov 16 '18 at 19:09
Théophile
19.5k12946
19.5k12946
asked Nov 16 '18 at 18:48
CreamPie
255
asked Nov 16 '18 at 18:48
CreamPie
255
asked Nov 16 '18 at 18:48
CreamPie
255
255
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
Every 6 terms are recurring, thus add first 6 terms and then the sum of first 99 terms is
(16x6)+Sum of three terms
Seems -12 is the answer.
answered Nov 20 '18 at 14:37
PiGuy
1487
add a comment |
HINT
We have that
$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$
answered Nov 16 '18 at 18:55
gimusi
1
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
|
show 3 more comments
If you compute the first $8$ terms, it will be evident that the sequence repeats in blocks of length $6$.
Summing the first $6$ terms, and then multiplying by $16$ yields the sum of the first $96$ terms.
To finish, add the three remaining terms (which are the same as the first three terms).
answered Nov 16 '18 at 19:09
quasi
36k22662
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
|
show 3 more comments
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
Every 6 terms are recurring, thus add first 6 terms and then the sum of first 99 terms is
(16x6)+Sum of three terms
Seems -12 is the answer.
answered Nov 20 '18 at 14:37
PiGuy
1487
add a comment |
Every 6 terms are recurring, thus add first 6 terms and then the sum of first 99 terms is
(16x6)+Sum of three terms
Seems -12 is the answer.
answered Nov 20 '18 at 14:37
PiGuy
1487
add a comment |
Every 6 terms are recurring, thus add first 6 terms and then the sum of first 99 terms is
(16x6)+Sum of three terms
Seems -12 is the answer.
answered Nov 20 '18 at 14:37
PiGuy
1487
Every 6 terms are recurring, thus add first 6 terms and then the sum of first 99 terms is
(16x6)+Sum of three terms
Seems -12 is the answer.
answered Nov 20 '18 at 14:37
PiGuy
1487
answered Nov 20 '18 at 14:37
PiGuy
1487
answered Nov 20 '18 at 14:37
PiGuy
1487
answered Nov 20 '18 at 14:37
PiGuy
1487
1487
add a comment |
add a comment |
HINT
We have that
$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$
answered Nov 16 '18 at 18:55
gimusi
1
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
|
show 3 more comments
HINT
We have that
$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$
answered Nov 16 '18 at 18:55
gimusi
1
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
|
show 3 more comments
HINT
We have that
$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$
answered Nov 16 '18 at 18:55
gimusi
1
HINT
We have that
$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$
answered Nov 16 '18 at 18:55
gimusi
1
edited Nov 16 '18 at 19:11
answered Nov 16 '18 at 18:55
gimusi
1
answered Nov 16 '18 at 18:55
gimusi
1
answered Nov 16 '18 at 18:55
gimusi
1
1
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
|
show 3 more comments
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
1
1
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
so how would I deduce the sums?
– CreamPie
Nov 16 '18 at 19:01
1
1
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 '18 at 19:05
1
1
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
I am feeling like a dork man
– CreamPie
Nov 16 '18 at 19:07
1
1
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 '18 at 19:07
2
2
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 '18 at 19:12
|
show 3 more comments
If you compute the first $8$ terms, it will be evident that the sequence repeats in blocks of length $6$.
Summing the first $6$ terms, and then multiplying by $16$ yields the sum of the first $96$ terms.
To finish, add the three remaining terms (which are the same as the first three terms).
answered Nov 16 '18 at 19:09
quasi
36k22662
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
|
show 3 more comments
If you compute the first $8$ terms, it will be evident that the sequence repeats in blocks of length $6$.
Summing the first $6$ terms, and then multiplying by $16$ yields the sum of the first $96$ terms.
To finish, add the three remaining terms (which are the same as the first three terms).
answered Nov 16 '18 at 19:09
quasi
36k22662
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
|
show 3 more comments
If you compute the first $8$ terms, it will be evident that the sequence repeats in blocks of length $6$.
Summing the first $6$ terms, and then multiplying by $16$ yields the sum of the first $96$ terms.
To finish, add the three remaining terms (which are the same as the first three terms).
answered Nov 16 '18 at 19:09
quasi
36k22662
If you compute the first $8$ terms, it will be evident that the sequence repeats in blocks of length $6$.
Summing the first $6$ terms, and then multiplying by $16$ yields the sum of the first $96$ terms.
To finish, add the three remaining terms (which are the same as the first three terms).
answered Nov 16 '18 at 19:09
quasi
36k22662
answered Nov 16 '18 at 19:09
quasi
36k22662
answered Nov 16 '18 at 19:09
quasi
36k22662
answered Nov 16 '18 at 19:09
quasi
36k22662
36k22662
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
|
show 3 more comments
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
1
1
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
It’s not a good idea in my opinion give full solution while the asker is trying to find it with some effort, you could just give the first part as a hint.
– gimusi
Nov 16 '18 at 19:15
1
1
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 '18 at 19:17
1
1
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@gimusi I am a 13 and half year boy getting efforts on higher maths ,gimusi and quasi I am on debt of you guys
– CreamPie
Nov 16 '18 at 19:20
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:21
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 '18 at 19:22
|
show 3 more comments
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