In a geometric series with first term a, common ratio r, is a=0 allowed? Or is that against the definition of a geometric series? Because everywhere I have searched, the only restriction in the definition of a geometric series is that r=/=0.
Now I say this because in C2, everyone (including the mark scheme) thinks it is ok to simply 'divide both sides of the equation by a", clearly ignoring the fact that they might be dividing by 0. I usually address the case where a=0 and say that if this is true, r can have any value besides 1 (this question is, for example, where they say that the sum to infinity of the series is 4 times the 3rd term). I know you might say that I should 'just assume a=/=0, but I don't consider that proper maths... Help me out?

PrimeLime
 Follow
 12 followers
 3 badges
 Send a private message to PrimeLime
 Thread Starter
Offline3ReputationRep: Follow
 1
 14052015 18:56

Revision help in partnership with Birmingham City University

lizard54142
 Follow
 7 followers
 3 badges
 Send a private message to lizard54142
Offline3ReputationRep: Follow
 2
 14052015 19:02
(Original post by PrimeLime)
In a geometric series with first term a, common ratio r, is a=0 allowed? Or is that against the definition of a geometric series? Because everywhere I have searched, the only restriction in the definition of a geometric series is that r=/=0.
Now I say this because in C2, everyone (including the mark scheme) thinks it is ok to simply 'divide both sides of the equation by a", clearly ignoring the fact that they might be dividing by 0. I usually address the case where a=0 and say that if this is true, r can have any value besides 1 (this question is, for example, where they say that the sum to infinity of the series is 4 times the 3rd term). I know you might say that I should 'just assume a=/=0, but I don't consider that proper maths... Help me out? 
Jai Sandhu
 Follow
 26 followers
 12 badges
 Send a private message to Jai Sandhu
Offline12ReputationRep: Follow
 3
 14052015 19:04
(Original post by PrimeLime)
In a geometric series with first term a, common ratio r, is a=0 allowed? Or is that against the definition of a geometric series? Because everywhere I have searched, the only restriction in the definition of a geometric series is that r=/=0.
Now I say this because in C2, everyone (including the mark scheme) thinks it is ok to simply 'divide both sides of the equation by a", clearly ignoring the fact that they might be dividing by 0. I usually address the case where a=0 and say that if this is true, r can have any value besides 1 (this question is, for example, where they say that the sum to infinity of the series is 4 times the 3rd term). I know you might say that I should 'just assume a=/=0, but I don't consider that proper maths... Help me out?(Original post by lizard54142)
Think about it. If a=0, what will the next term be... a*r = 0*r = 0 
lizard54142
 Follow
 7 followers
 3 badges
 Send a private message to lizard54142
Offline3ReputationRep: Follow
 4
 14052015 19:06
(Original post by Jai Sandhu)
There is no such thing as a geometric sequence with 0 as the first term. 
Jai Sandhu
 Follow
 26 followers
 12 badges
 Send a private message to Jai Sandhu
Offline12ReputationRep: Follow
 5
 14052015 19:06
(Original post by lizard54142)
That is the point I'm trying to make. 
 Follow
 6
 14052015 19:07
(Original post by Jai Sandhu)
There is no such thing as a geometric sequence with 0 as the first term.
The sequence: 0,0,0,0,0,...
is still a geometric sequence, so it does exist. 
lizard54142
 Follow
 7 followers
 3 badges
 Send a private message to lizard54142
Offline3ReputationRep: Follow
 7
 14052015 19:09
(Original post by Jai Sandhu)
I know, I thought I would reiterate it
(Original post by Xin Xang)
Thats not true.
The sequence: 0,0,0,0,0,...
is still a geometric sequence, so it does exist. 
Jai Sandhu
 Follow
 26 followers
 12 badges
 Send a private message to Jai Sandhu
Offline12ReputationRep: Follow
 8
 14052015 19:13

 Follow
 9
 14052015 19:13
Let a=0 so that the sequence is r(0),r^2(0),... and so on. 
Kevin De Bruyne
 Follow
 615 followers
 21 badges
 Send a private message to Kevin De Bruyne
 Very Important Poster
Online21ReputationRep:Very Important Poster Follow
 10
 14052015 19:14
Would you mind ponting to a past paper/example? There might be something in the question.

 Follow
 11
 14052015 19:18
(Original post by Jai Sandhu)
Not only this but for a geometric series to converge when you sum to infinity by definition r < 1. Yet, that series you suggests can have have any value for r, including those greater than r yet the sum to infinity will be 0 and not diverge even though r may be greater than 1. 
lizard54142
 Follow
 7 followers
 3 badges
 Send a private message to lizard54142
Offline3ReputationRep: Follow
 12
 14052015 19:22
(Original post by Xin Xang)
r could be a non zero real number.
Let a=0 so that the sequence is r(0),r^2(0),... and so on.
So for your series:
?Last edited by lizard54142; 14052015 at 19:24. 
PrimeLime
 Follow
 12 followers
 3 badges
 Send a private message to PrimeLime
 Thread Starter
Offline3ReputationRep: Follow
 13
 14052015 19:25
(Original post by SeanFM)
Would you mind ponting to a past paper/example? There might be something in the question. 
 Follow
 14
 14052015 19:30
(Original post by lizard54142)
I don't consider that to be geometric. The ratio between consecutive terms is defined as:
So for your series:
?
So technically there is still a ratio. We just wouldn't know what it was from the sequence. 
PrimeLime
 Follow
 12 followers
 3 badges
 Send a private message to PrimeLime
 Thread Starter
Offline3ReputationRep: Follow
 15
 14052015 19:31
(Original post by lizard54142)
I don't consider that to be geometric. The ratio between consecutive terms is defined as:
So for your series:
?
But I still don't know why they don't specify a=/=0 in the formal definition.Last edited by PrimeLime; 14052015 at 19:32. 
PrimeLime
 Follow
 12 followers
 3 badges
 Send a private message to PrimeLime
 Thread Starter
Offline3ReputationRep: Follow
 16
 14052015 19:33
(Original post by Xin Xang)
Which is an indeterminate form. We wouldn't necessarily know what the value of r is, but that doesn't mean it doesn't exist.
So technically there is still a ratio. We just wouldn't know what it was from the sequence. 
Jai Sandhu
 Follow
 26 followers
 12 badges
 Send a private message to Jai Sandhu
Offline12ReputationRep: Follow
 17
 14052015 19:35
(Original post by PrimeLime)
Well then there's this argument, which is pretty convincing. Argh!! 
 Follow
 18
 14052015 19:39
(Original post by Jai Sandhu)
That argument is more convincing for saying it is not a geometry series. 
lizard54142
 Follow
 7 followers
 3 badges
 Send a private message to lizard54142
Offline3ReputationRep: Follow
 19
 14052015 19:41
(Original post by Xin Xang)
Which is an indeterminate form. We wouldn't necessarily know what the value of r is, but that doesn't mean it doesn't exist.
So technically there is still a ratio. We just wouldn't know what it was from the sequence.
I understand your reasoning, but I am still unconvinced as you can't get away from the issue of dividing by zero. Interesting discussion though
EDIT: 0, 0, 0 ... is arithmetic in my eyes.Last edited by lizard54142; 14052015 at 19:43. 
PrimeLime
 Follow
 12 followers
 3 badges
 Send a private message to PrimeLime
 Thread Starter
Offline3ReputationRep: Follow
 20
 14052015 19:45
(Original post by lizard54142)
So you have an "undefined" ratio say. But in order to define a geometric series, you need to know "a", and "r". If you are given these values you can generate any geometric sequence, these values are what defines the sequence. How would you define the sequence 0, 0, 0 .... ? You would define it differently from every other geometric sequence, because r could be any real number. So is it still geometric?
I understand your reasoning, but I am still unconvinced as you can't get away from the issue of dividing by zero. Interesting discussion though
EDIT: 0, 0, 0 ... is arithmetic in my eyes.
 1
 2
Related discussions
 C2 simple geometric series question
 AQA C2 Geometric series question
 Really stuck on a geometric series question
 C2: Geometric Series Question
 Geometric series question
 Geometric Series Question Help
 C2 geometric series question
 Maths geometric series question
 Help with Geometric Series question
 having difficulties with Geometric series question
Related university courses

Mathematics
University of Birmingham

Financial Mathematics
University of Hertfordshire

Mathematics and Physics
University of Dundee

Mathematics for Finance and Management
University of Portsmouth

Mathematics
University of Winchester

Mathematics
University of Oxford

Mathematics (Including Year Abroad)
University of Essex

Mathematics and Statistics (including placement year)
University of Essex

Mathematics with Foundation
Durham University

Pure Mathematics (Fast Track)
University of St Andrews
We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.
 SherlockHolmes
 Notnek
 charco
 Mr M
 Changing Skies
 F1's Finest
 rayquaza17
 RDKGames
 davros
 Gingerbread101
 Kvothe the Arcane
 TeeEff
 The Empire Odyssey
 Protostar
 TheConfusedMedic
 nisha.sri
 claireestelle
 Doonesbury
 furryface12
 Amefish
 harryleavey
 Lemur14
 brainzistheword
 Rexar
 Sonechka
 TheAnxiousSloth
 EstelOfTheEyrie
 CoffeeAndPolitics
 an_atheist
 Labrador99
 EmilySarah00
 thekidwhogames
 entertainmyfaith
 Eimmanuel