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Three successive terms of a G.P. will fo...

Three successive terms of a G.P. will form the sides of a triangle if the common ratio r satisfies the inequality

A

`(sqrt(3)-1)/(2)ltrlt(sqrt(3)+1)/(2)`

B

`(sqrt(5)-1)/(2)ltrlt(sqrt(5)+1)/(2)`

C

`(sqrt(2)-1)/(2)ltrlt(sqrt(2)+1)/(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Let the lengths of the sides of the triangle be `a,ar,ar^(2)`. We have the following three cases :
CASE I When r=1
In this case, the lengths of sides of the triangle are a,a,a i.e. the triangle is equilateral.
CASE II When `rgt1`
In this case, the length of the largest side is `ar^(2)`. Therefore, the triangle will be formed, if
`a+argtar^(2)`
`rArr" "r^(2)-r-1lt0`
`rArr" "(1-sqrt(5))/(2)ltrlt(1+sqrt(5))/(2)`
`rArr" "rlt(1+sqrt(5))/(2)" "[becausergt1]` . . ..(i)
CASE III When `rlt1`
In this case, the length of the largest side is a. So, the triangle will be formed, if
`ar+ar^(2)gta`
`rArr" "r^(2)+r-1gt0`
`rArr" "rlt(-1-sqrt(5))/(2)or,rgt(-1+sqrt(5))/(2)`
`rArr" "(sqrt(5)-1)/(2)ltrlt1" "[becauserlt1]` . . . (ii)
From (i) and (ii), we obtain : `(sqrt(5)-1)/(2)ltrlt(sqrt(5)+1)/(2)`
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