Latest Elitmus Aptitude Question SOLUTION: a,b,c are positive numbers such that they are in increasing Geometric progression then how many such numbers are there in (loga+logb+logc)/6 =log6

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(#M40157508) ELITMUS Numerical Ability question quants Keep an EYE

a,b,c are positive numbers such that they are in increasing Geometric progression then how many such numbers are there in (loga+logb+logc)/6 =log6

Asked In Elitmus SHUBHAM (7 years ago)

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Subrata Dutta  (7 years ago)

given that a,b,c are in gp
so b/a=c/b =>b^2=ac
(loga+logb+logc)/6=log6
=>log(abc)=6*log6
=>log(ac*b)=log(6^6)
=>log(b^3)=log(6^6)
=>b^3=6^6=>b^3=(6^2)^3
=>b=6^2=36
it means a=2,c=18 or a=3 c=12 or a=4 c=9 or a=6 c=6.
hence there are four such numbers

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akshay  (7 years ago)

(loga+logb+logc)/6=log6
or, log(abc)=6*log6
or, log(ac*b)=log(6^6)
or, log(b^3)=log(6^6)
or, b^3=6^6
or, b^3=(36)^3
or, b=6^2
or, b=36
now we got b=36 and because a,b,c are in a G.P
b^2=a*c
that means, a*c=6^4.
so, there are 5 possible values for a,b & c which are:-
(2,36,648) (3,36,432) (6,36,216) (12,36,108) (18,36,72).
and also a,b,c can not be (36,36,36) because question already told us that GP is incresing.

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priyadarshan kumar  (7 years ago)

in question if it is positive number then infinite number of solution possible but if it is positive integer then 8 solutions are possible

a = a
b = ar
c = ar^2

from question

log a + log b + log c = 3log(ar)
solving the equation given in question we get

log(ar) = log36
hence ar = 36

so the gp is

1, 36, 1296
2, 36, 648
3, 36, 432
4, 36, 324
6, 36, 216
8, 36, 162
9, 36, 144
12, 36, 108
18, 36, 72

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