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.

- anonymous

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- anonymous

@hartnn

- hartnn

if x,y,z form an geometric sequence,
then \(y^2 =xz\)
so, what about
a,-4 and b ?

- anonymous

ab=(-4)^2=16
why?

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## More answers

- anonymous

it is also fixed?

- hartnn

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)

- anonymous

okay, i understand

- hartnn

or in other words, they form a proportion, \(\dfrac{1st \: term}{2nd \: term}=\dfrac{2nd \: term}{3rd\: term}\)

- hartnn

16 is correct :)
now if the terms are in arithmetic sequence,
then 2* 2nd term = 1st term + 3rd term
use this...

- anonymous

wow, i like the formulae

- hartnn

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) = a-b
you'll get same equation :)

- anonymous

yup, got it

- hartnn

tell me what u get for a and b when you get it....2 equations, 2 unknowns...

- anonymous

-4+d=b
b+d=a
ar=-4
ar^2=b
are the equations correct?

- hartnn

ar = -4 ? how ?

- anonymous

i don't know, i just guess.....it's written couple of days ago. i forgot how...

- hartnn

forget about the 'd' and 'r' now
we have 2 equations in a and b
ab = 16
and from b-(-4) = a-b, we have 2b =a-4
you got these ? how..

- anonymous

uh, you said this moment ago. got it.
a=8 , b=2

- hartnn

those values are correct.
so b) is done right ?
now c) part
can you find the common ratio in 8,-4,2,....?

- anonymous

*-2

- hartnn

b=2 was correct....

- anonymous

common ratio is -2

- hartnn

ohh! yes, r= -2 :)
first term = a1 = 8
SUm to infinite is just \(\large \dfrac{a_1}{r-1}\)

- anonymous

what? -8/3?

- hartnn

ohh....the r is not -2 :P

- anonymous

-1/2 ?

- hartnn

yes.
r = 2nd term / 1st term = 3rd trm/ 2nd term =.... = -4/8 = -1/2

- anonymous

-16/3 ...

- hartnn

since r is LESS than 1, we use
\(\huge S_\infty = \dfrac{a_1}{1-r}\)

- anonymous

lol

- hartnn

yes, there are 2 formula, depending on r less than 1 or greater than 1

- anonymous

i will remember. I just forgot at that moment..

- hartnn

try the last aprt ?

- anonymous

i hate doing +ve / -ve ....

- hartnn

remove all the negatives...
8,-4,2,-1,1/2 ....
to
8,2,1/2 ....
now whats the common ratio r =... ?

- anonymous

oh, i know what's the world happening!
r=1/4

- hartnn

yessss!

- hartnn

and since 1/4 is less than 1
you use
a/ (1-r)

- anonymous

32/3

- anonymous

million thanks

- hartnn

\(\huge \color{red}\checkmark\)

- hartnn

welcome ^_^

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