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#1 2013-06-19 12:15:58

Agnishom
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Glamor Geometric Series

Integers a, b, c, d and e satisfy 50<a<b<c<d<e<500, and a,b,c,d,e form a geometric sequence. What is the sum of all possible distinct values of a?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#2 2013-06-19 13:53:49

bobbym
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Re: Glamor Geometric Series

Hi;

[removed for security reasons]


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#3 2013-06-19 15:01:14

Agnishom
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Re: Glamor Geometric Series

That seems to be correct!

But how isn't a = 50*(10^(1/6))


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#4 2013-06-19 15:56:52

bobbym
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Re: Glamor Geometric Series

[removed by administrator]

Post cleared.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#5 2013-06-19 17:59:47

bob bundy
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Re: Glamor Geometric Series

hi

I just reasoned it out from the integer property.  Useful to have a spreadsheet but, I suppose I could have worked these numbers out anyway.

Bob


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You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#6 2013-06-19 22:12:45

Agnishom
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Re: Glamor Geometric Series

Help! I did not understand anything at all sad

BTW This is a Level 4 problem


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#7 2013-06-19 22:25:45

bobbym
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Re: Glamor Geometric Series

Hi;

[removed by administrator]


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#8 2013-06-19 22:49:44

bob bundy
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Re: Glamor Geometric Series

hi Agnishom,

Are you wanting an outline of my answer?  (note; it was mainly trial and improvement)?

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#9 2013-06-19 22:54:17

Agnishom
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Re: Glamor Geometric Series

Yep and please explain it. I don't understand it at all


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#10 2013-06-19 23:09:18

bob bundy
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Re: Glamor Geometric Series

hi Agnishom,

I made a spreadsheet just to make the calculations easy.

r is the common ratio.

r>1 or you won't get b>a etc.

I tried r = 2 and even with a = 51 the numbers quickly get out of range, so

1 < r < 2

So I tried r = 1.1

The numbers become decimals not integers.

Tried r = 1.5

Term will be the previous term plus half of it so 'a' needs to be even.

Then I realised that unless a is a multiple of 16, the successive halving will generate a decimal.

eg If a = 100 then b = 150 and c = 225 and then we get decimals

So I tried 64 and got an answer, then 80, then 96 and then I was out of range.

So what else might work.

r = 1.25  No because successive quartering means I have to start with a large power of 4, out of range.

r = 1.333333   Only if I start with a multiple of 3^4

That gave me one more solution.

r = 1.666666 goes out of range.

Any other (1 plus fraction) won't work  because

say r = 1 + 1/n

a must be a fourth power of a multiple of n.  Out of range for all n > 3

And that's about it.

As you can see I sort of discovered what to do as I tried numbers and gradually developed a rule.

smile

Bob

got to go out so any clarification will have to wait 'til later.


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#11 2013-06-20 00:59:24

Agnishom
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Re: Glamor Geometric Series

Just passed like a tangent over my tiny brain. After my examination stress is reduced, I'll trying reading that again


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#12 2013-06-20 01:03:58

bobbym
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Re: Glamor Geometric Series

Did you try programming it?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#13 2013-06-20 01:42:46

Agnishom
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Re: Glamor Geometric Series

No.

I am not at all sure what the problem means.
Doesn't it mean that a = 50x, b = 50x*x, c = 50x*x*x, d = 50x*x*x*x, e = 50x*x*x*x*x
The ratio needs to be fixed, right?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#14 2013-06-20 01:45:49

bobbym
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Re: Glamor Geometric Series

Yes, exactly!


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#15 2013-06-20 01:51:02

phanthanhtom
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Re: Glamor Geometric Series

a min = 51.
e max = 499.
==> ratio max ~ 1.768
Now if ratio was expressed m/n then a is divisible by n^4 for e to be integral. Therefore n < 5, if n > 4 => a > 495 contradiction.
Now try ratios m/n = 3/2; 4/3; 5/3; 5/4; 7/4.
E.g. m/n = 3/2. ==> e = 81a/16 < 500
==> a < 99. But a is divisible by 2^4 therefore 3 values 64, 80 and 96.

#16 2013-06-20 01:56:52

bobbym
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Re: Glamor Geometric Series

There is one more.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#17 2013-06-20 01:57:50

Agnishom
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Re: Glamor Geometric Series

So 500 = 50x^6 => 10 = x^6

x has exactly 6 solutions the sum of them is zero.

Since, a = 50x, the sum of all a should be zero too.
How can it be an integer?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#18 2013-06-20 02:42:38

bob bundy
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Re: Glamor Geometric Series

hi Agnishom,

You are still misunderstanding the question.

The number 'r' is called the common ratio of the geometric series.  This means that

a is the first term

b = ar is the second

c = ar^2 is the third

d = ar^3 is the fourth

e = ar^4 is the fifth (and, for our purposes, the last).

Here's what I tried first.

r = 1.1 a = 51  gives b = 56.1  c = 61.71  d = 67.881  e = 74.6891

That won't do because they are supposed to be integers.

So I tried r = 1.5 a = 62

Better because b is also now an integer but c isn't.

So I started thinking ...... dizzy

For e to be an integer, d must be even. (1.5 x even is bound to be an integer)

So c must be a multiple of 4

eg. c = 84 => d =   126 => e = 189

Reasoning backwards, b must be a multiple of 8 and a must be a multiple of 16

Tried r = 1.5 a = 64  => b = 96 => c = 144  => d = 216 => e = 324

Success!

So keep r = 1.5 and try other multiples of 16

80,120,180,270,405
96,144,216,324,486

This last is only just under 500 so there's no point trying any more.

Now what else could r be ?

r = 1 and a third ??

If any term is multiplied by r you'd get that term + 1/3 of that term

So for e to be an integer, d must be divisible by 3 => c must be divisible by 9 => b must be divisible by 27 => a must be divisible by 81

So try 81,108,144,192,256

Could r be 1 and two thirds.

You get integers but it goes above 500.

Could r be 1 and a quarter ?

For e to be an integer, d must be divisible by 4 => c by 16, => b by 64 => a by 256

It'll work but be out of range.  Same goes for any other r of the form 1 + a proper fraction.

Could r be irrational ?

No.  Fails at the first hurdle because integer x irrational gives an irrational and not an integer.

So that's the complete list.

Sum of a = 64 + 80 + 96 + 81 = 321

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#19 2013-06-20 12:55:36

Agnishom
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Re: Glamor Geometric Series

Then I realised that unless a is a multiple of 16, the successive halving will generate a decimal.

Why and How?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#20 2013-06-20 15:08:26

phanthanhtom
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Re: Glamor Geometric Series

a has to be divisible by 16 so that it could cancel with 16 in the denominator.

#21 2013-06-20 17:16:34

bob bundy
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Re: Glamor Geometric Series

hi Agnishom,

Let's take as an example the sequence that ends e = 405

That required that d was even.  And it was d = 270.

But if c is 'just' even it will mean that d is an integer, but it won't guarantee that e is an integer.

From c to e is



Similarly



and finally



so we have c = 180.  This is divisible by 4

and b = 120.  This is divisible by 8

and a = 80.  This is divisible by 16

Try out some other numbers and you'll begin to see why it has to be 16.

(and for r = 1 and a third a has to be divisible by 3 x 3 x 3 x 3 .)

Bob


You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

#22 2013-07-03 02:47:14

Agnishom
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Re: Glamor Geometric Series

The Featured Solution: https://brilliant.org/i/b6ozMJ/


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#23 2013-07-03 02:59:37

bobbym
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Re: Glamor Geometric Series

Yes, if you are signed in as a member.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#24 2013-07-03 03:10:27

Agnishom
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Re: Glamor Geometric Series

New Problem There are 5 kinds of dishes i, ii, iii, iv, and v. Four Customers A, B, C and D choose a dish at random. What is the probability that person D chooses a previously unordered dish?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#25 2013-07-03 03:16:58

bobbym
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Re: Glamor Geometric Series

Although I have the exact answer by experimental methods I am still working on a good solid method that would please an olympic dude.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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