CTK Exchange Front Page Movie shortcuts Personal info Awards Reciprocal links Terms of use Privacy Policy Cut The Knot! MSET99 Talk Games & Puzzles Arithmetic/Algebra Geometry Probability Eye Opener Analog Gadgets Inventor's Paradox Did you know?... Proofs Math as Language Things Impossible My Logo Math Poll Other Math sit's Guest book News sit's Recommend this site |Store|       CTK Exchange

 Subject: "Hard Sequence to crack!!" Previous Topic | Next Topic
 Conferences The CTK Exchange High school Topic #280 Printer-friendly copy Email this topic to a friend 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 okthanx!

Graham C
Member since Feb-5-03
Apr-23-04, 07:03 AM (EST)    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, leaving1/10 + 2/100 + 3/1000....That equals 1/9 + 1/100 + 2 /1000... which then equals1/9 + 1/90 + 1/1000 + 2/10000....which equals1/9 + 1/90 + 1/900 + 1/10000... and so onWhich gives you 1/9 + 1/90 + 1/900....which equals1/9 * (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 have2/3 + 10/81 = (54+10)/81 = 64/81 = (8/9)^2.Given the neat answer there may be a more elegant way of doing it. 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^nis 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.

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

 Conferences | Forums | Topics | Previous Topic | Next Topic
 Select another forum or conference Lobby The CTK Exchange (Conference)   |--Early math (Public)   |--Middle school (Public)   |--High school (Public)   |--College math (Public)   |--This and that (Public)   |--Guest book (Public) You may be curious to have a look at the old CTK Exchange archive.  