Finding sum of terms in a sequence

up vote
2
down vote

favorite

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...

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 at 18:48

CreamPie

255

add a comment |

up vote
2
down vote

favorite

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...

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 at 18:48

CreamPie

255

add a comment |

up vote
2
down vote

favorite

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...

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 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

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 at 18:48

CreamPie

255

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 at 18:48

CreamPie

255

edited Nov 16 at 19:09

Théophile

19.3k12946

edited Nov 16 at 19:09

Théophile

19.3k12946

edited Nov 16 at 19:09

Théophile

19.3k12946

asked Nov 16 at 18:48

CreamPie

255

asked Nov 16 at 18:48

CreamPie

255

asked Nov 16 at 18:48

CreamPie

255

add a comment |

3 Answers
3

active

oldest

votes

up vote
1
down vote

accepted

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 at 14:37

PiGuy

1487

add a comment |

up vote
2
down vote

HINT

We have that

$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

1

so how would I deduce the sums?
– CreamPie
Nov 16 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 at 19:05

1

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

1

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 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 at 19:12

|
show 3 more comments

up vote
1
down vote

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 at 19:09

quasi

35.9k22562

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 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 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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:22

|
show 3 more comments

Your Answer

StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\$","\$"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});

}
});

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Post as a guest

Name

Required, but never shown

StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3001505%2ffinding-sum-of-terms-in-a-sequence%23new-answer', 'question_page');
}
);

Post as a guest

Name

Required, but never shown

3 Answers
3

active

oldest

votes

3 Answers
3

active

oldest

votes

up vote
1
down vote

accepted

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 at 14:37

PiGuy

1487

add a comment |

up vote
1
down vote

accepted

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 at 14:37

PiGuy

1487

add a comment |

up vote
1
down vote

accepted

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 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 at 14:37

PiGuy

1487

answered Nov 20 at 14:37

PiGuy

1487

answered Nov 20 at 14:37

PiGuy

1487

answered Nov 20 at 14:37

PiGuy

1487

add a comment |

up vote
2
down vote

HINT

We have that

$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

1

so how would I deduce the sums?
– CreamPie
Nov 16 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 at 19:05

1

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

1

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 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 at 19:12

|
show 3 more comments

up vote
2
down vote

HINT

We have that

$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

1

so how would I deduce the sums?
– CreamPie
Nov 16 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 at 19:05

1

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

1

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 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 at 19:12

|
show 3 more comments

up vote
2
down vote

HINT

We have that

$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

HINT

We have that

$$a_{n}= frac {a_{n-1}}{a_{n-2}}=frac {a_{n-2}}{a_{n-3}}frac {1}{a_{n-2}}=frac {1}{a_{n-3}}=ldots=a_{n-6}$$

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

edited Nov 16 at 19:11

answered Nov 16 at 18:55

gimusi

87.7k74393

answered Nov 16 at 18:55

gimusi

87.7k74393

answered Nov 16 at 18:55

gimusi

87.7k74393

1

so how would I deduce the sums?
– CreamPie
Nov 16 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 at 19:05

1

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

1

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 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 at 19:12

|
show 3 more comments

1

so how would I deduce the sums?
– CreamPie
Nov 16 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 at 19:05

1

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

1

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 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 at 19:12

so how would I deduce the sums?
– CreamPie
Nov 16 at 19:01

Take another step, what can we deduce? If $a_n=1/a_{n-3}$ what is $a_{n-3}$ equal to?
– gimusi
Nov 16 at 19:05

I am feeling like a dork man
– CreamPie
Nov 16 at 19:07

@CreamPie Did you calculate any values beyond $a_5$?
– Théophile
Nov 16 at 19:07

You are almost near, you don’t need more help than that. Take the pleasure to find it by your own.
– gimusi
Nov 16 at 19:12

|
show 3 more comments

up vote
1
down vote

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 at 19:09

quasi

35.9k22562

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 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 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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:22

|
show 3 more comments

up vote
1
down vote

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 at 19:09

quasi

35.9k22562

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 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 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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:22

|
show 3 more comments

up vote
1
down vote

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 at 19:09

quasi

35.9k22562

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 at 19:09

quasi

35.9k22562

answered Nov 16 at 19:09

quasi

35.9k22562

answered Nov 16 at 19:09

quasi

35.9k22562

answered Nov 16 at 19:09

quasi

35.9k22562

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 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 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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 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 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 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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:22

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 at 19:15

I made a judgement that in this case, more was needed. And also less (more pattern based, less symbolic).
– quasi
Nov 16 at 19:17

@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 at 19:20

@quasi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:21

@gimusi can you recommend a book for me to study these things?
– CreamPie
Nov 16 at 19:22

|
show 3 more comments

draft saved

draft discarded

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Post as a guest

Name

Required, but never shown

Post as a guest

Name

Required, but never shown

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Post as a guest

Name

Required, but never shown

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Post as a guest

Name

Required, but never shown

Sign up or log in

StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});

Post as a guest

Name

Required, but never shown

Name

Required, but never shown

Name

Required, but never shown

This page is only for reference, If you need detailed information, please check here

搜尋此網誌

Vrftjkry