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a and b are +ve integers. a,4,b form a geometric sequence, and 4,b,a form an arithmetric sequence.
(a) Find the value of ab.
(b) Find the values of a and b.
(c) (i) Find the sum to infinity of the geometric sequence a,4,b,... .
(ii) Find the sum to infinity of all the terms that are +ve in the geometric sequence a,4,b.
 9 months ago
 9 months ago
a and b are +ve integers. a,4,b form a geometric sequence, and 4,b,a form an arithmetric sequence. (a) Find the value of ab. (b) Find the values of a and b. (c) (i) Find the sum to infinity of the geometric sequence a,4,b,... . (ii) Find the sum to infinity of all the terms that are +ve in the geometric sequence a,4,b.
 9 months ago
 9 months ago

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hartnnBest ResponseYou've already chosen the best response.1
if x,y,z form an geometric sequence, then \(y^2 =xz\) so, what about a,4 and b ?
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
i can get you a simple explanation here... each next term of geometric sequence is MULTIPLIED by the common ratio 'r' so, in general, terms are a,ar,ar^2,ar^3..... when we have just 3 terms, we have a, ar, ar^2 for this (1st term * 3rd term) = (2nd term)^2 (you can check)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
or in other words, they form a proportion, \(\dfrac{1st \: term}{2nd \: term}=\dfrac{2nd \: term}{3rd\: term}\)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
16 is correct :) now if the terms are in arithmetic sequence, then 2* 2nd term = 1st term + 3rd term use this...
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
wow, i like the formulae
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
or if you don't want to remember the formula, you should know that difference between the terms is constant in arithmetic sequence so, difference between b and 4 = difference between a and b b  (4) = ab you'll get same equation :)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
tell me what u get for a and b when you get it....2 equations, 2 unknowns...
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
4+d=b b+d=a ar=4 ar^2=b are the equations correct?
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
i don't know, i just guess.....it's written couple of days ago. i forgot how...
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
forget about the 'd' and 'r' now we have 2 equations in a and b ab = 16 and from b(4) = ab, we have 2b =a4 you got these ? how..
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
uh, you said this moment ago. got it. a=8 , b=2
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
those values are correct. so b) is done right ? now c) part can you find the common ratio in 8,4,2,....?
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
ohh! yes, r= 2 :) first term = a1 = 8 SUm to infinite is just \(\large \dfrac{a_1}{r1}\)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
ohh....the r is not 2 :P
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
yes. r = 2nd term / 1st term = 3rd trm/ 2nd term =.... = 4/8 = 1/2
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
since r is LESS than 1, we use \(\huge S_\infty = \dfrac{a_1}{1r}\)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
yes, there are 2 formula, depending on r less than 1 or greater than 1
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
i will remember. I just forgot at that moment..
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
i hate doing +ve / ve ....
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
remove all the negatives... 8,4,2,1,1/2 .... to 8,2,1/2 .... now whats the common ratio r =... ?
 9 months ago

kryton1212Best ResponseYou've already chosen the best response.1
oh, i know what's the world happening! r=1/4
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
and since 1/4 is less than 1 you use a/ (1r)
 9 months ago

hartnnBest ResponseYou've already chosen the best response.1
\(\huge \color{red}\checkmark\)
 9 months ago
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