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Subject: "Hard Sequence to crack!!"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #280
Reading Topic #280
A-9585
guest
Apr-22-04, 07:01 PM (EST)
 
"Hard Sequence to crack!!"
 
   Does anyone know the sum of this infinite sequence?
7/10 + 8/100 + 9/1000 + 10/10000 ...

I know the formula of it is (6 + n)/(10^n), but i dont know how to find the infinite sum of the sequence.

any ideas are ok
thanx!


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Graham C
Member since Feb-5-03
Apr-23-04, 07:03 AM (EST)
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1. "RE: Hard Sequence to crack!!"
In response to message #0
 
   >Does anyone know the sum of this infinite sequence?
>7/10 + 8/100 + 9/1000 + 10/10000 ...
>
>I know the formula of it is (6 + n)/(10^n), but i dont know
>how to find the infinite sum of the sequence.
>
>any ideas are ok
>thanx!

I'll assume you're aware that the sum of 1/10 + 1/100 + 1/1000... = 1/9 (if not, just think of 0.11111.....)

6/10 + 6/100 + 6/1000... therefore = 6/9 = 2/3. Deduct that from the series you started with, leaving
1/10 + 2/100 + 3/1000....

That equals 1/9 + 1/100 + 2 /1000... which then equals
1/9 + 1/90 + 1/1000 + 2/10000....which equals
1/9 + 1/90 + 1/900 + 1/10000... and so on

Which gives you 1/9 + 1/90 + 1/900....which equals
1/9 * (1 + 1/10 + 1/100...) = 1/9*(1+1/9) = 10/81.

Add back the 2/3 we took out at the beginning and you have
2/3 + 10/81 = (54+10)/81 = 64/81 = (8/9)^2.

Given the neat answer there may be a more elegant way of doing it.


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Mauer-Oats
guest
Apr-26-04, 01:09 PM (EST)
 
2. "RE: Hard Sequence to crack!!"
In response to message #1
 
   One "more elegant" way is this...
The derivative of the geometric series
Sum_{n>0} r^n
is
Sum_{n>0} n r^{n-1}

Take the derivative of both sides of the formula you know for the sum of a geometric series and work from there.


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Irving
guest
Apr-28-04, 06:59 AM (EST)
 
3. "RE: Hard Sequence to crack!!"
In response to message #0
 
   Let N = 7/10 + 8/100 + 9/1000 + ...

Then 10N = 7 + 8/10 + 9/100 + ...

Subtracting N from 10N, we get 9N.

This can also be expressed as follows:
(7 + 8/10 + 9/100 + ...)-(7/10 + 8/100 + 9/1000 + ...) =
7 + 1/10 + 1/100 + 1/1000 + 1/10,000 + ... =
7 + 1/9 =
64/9

Thus, 9N = 64/9.
Divide each side by 9: N = 64/81


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