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 one year ago
The general term of an infinite geometric sequence is given by:
T(n)=6[(x+1)/8)^(n)
(a) Find the range of the values of x for which the sum to infinity exists.
(b) Find the sum to infinity of the sequence in terms of x.
(c) If the sum to infinity of the sequence is 18, find the value of x.
 one year ago
The general term of an infinite geometric sequence is given by: T(n)=6[(x+1)/8)^(n) (a) Find the range of the values of x for which the sum to infinity exists. (b) Find the sum to infinity of the sequence in terms of x. (c) If the sum to infinity of the sequence is 18, find the value of x.

This Question is Closed

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0\[T(n)=6(\frac{ x+1 }{ 8 })^{n}\]

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3the sum to infinity will converge(exist) if \(r<1\) can you find 'r' from your general term ?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3general equation, \(T(n)=a_1r^{n1}\)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3find a1 and 'r' both from your general term, you will need both

primeralph
 one year ago
Best ResponseYou've already chosen the best response.0dw:1372921816382:dw

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3compare, try to make the power as (n1) \(\large 6 (\dfrac{x+1}{8})^n = 6 \dfrac{x+1}{8}(\dfrac{x+1}{8})^{n1}\) now compare this with a1 r^{n1} find a1 and r

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3then for the range of values of x, just do r <1

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3ohh..because @primeralph wrote "and" there! its actually "or"

primeralph
 one year ago
Best ResponseYou've already chosen the best response.0dw:1372923436567:dw

primeralph
 one year ago
Best ResponseYou've already chosen the best response.0@Jhannybean The x is only a part of r.

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1Oh nevermind.... I was looking at it the wrong way.

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0i am still confusing..

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1No i wasn't referring to that.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3@kryton1212 where are you confused ?

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3you got this much ? \(\large \dfrac{x+1}{8}<1\) ?

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0everywhere... i still cannot get what you mean..

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0what does  this line mean

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3the absolute value sign

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3\(if \quad a<b \\ a<b \quad or \quad a>b\)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3so, what about (x+1)/8 <1 ?

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1OH its an OR. OK that makes more sense now, Got it.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3\(if \quad a<b \\ a<b \quad or \quad a>b\) \(\large if \quad \dfrac{x+1}{8}<1 \\ \large \dfrac{x+1}{8}<1 \quad or \quad \dfrac{x+1}{8}>1\)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3now can you get the range ?

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1So just solve for two xvalues,and see which xvalue fits the description of r<1.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3yes, correct, thats your 1st part :)

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0thanks:) i got it ;D and part b.

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3sum to infinity = \(\large \dfrac{a_1}{1r}\)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3\(\large a_1 =\dfrac{6(x+1)}{8}\)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3could you find the sum to infinity ?

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0dw:1372925264670:dw

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3The general term in an geometric series is \(T(n)=a_1r^{n1}\) where, a1 is the 1st term of the series (which we get when we put n=1 in the general term) and 'r' is the common ratio, (ratio of next term to current term)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3if you want more details about geometric sequence, you can go through this when u have time, http://openstudy.com/study#/updates/503bb2a0e4b007f9003103b0

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3any more doubts ? what u got as sum to infinity ?

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0dw:1372925634484:dw

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1Yay!! now simplify :)

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3correct! now for the last part, the sum is given as 18 so, getting x from 6 (x+1) / (7x) = 18 is just simple algebra :)

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1\[\large 6 (\dfrac{x+1}{8})^n = 6 \dfrac{x+1}{8}(\dfrac{x+1}{8})^{n1}\]\[\large a_{1}=\cfrac{6(x+1)}{8} \ \text{and} \ r = \left(\frac{x+1}{8}\right)\]\[\large s_n= \cfrac{ \cfrac{6(x+1)}{8}}{\cfrac{8(x+1)}{8}}\]

hartnn
 one year ago
Best ResponseYou've already chosen the best response.3to give you a head start, multiply both sides by 7x

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1\[\large s_{n} = \cfrac{3(x+1)}{4}\cdot \cfrac{8}{x+7} = \frac{6(x+1)}{x+7}\]

kryton1212
 one year ago
Best ResponseYou've already chosen the best response.0thanks @hartnn and @Jhannybean :)

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.1So yeah...you equal that equation to 18 and solve for x :) You stated that x=5, let's check that! \[\large 18 = \cfrac{6(5+1)}{5+7}\]\[\large 18=18 \ \checkmark \] good job!

hartnn
 one year ago
Best ResponseYou've already chosen the best response.36x+6 = 12618x 24x = 120 x=5 is correct :)
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