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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Jhannybean
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.3yes, correct, thats your 1st part :)

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

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

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

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

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

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

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

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

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

Jhannybean
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.0thanks @hartnn and @Jhannybean :)

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