Question

The number q is a quotient of a geometric sequence (with positive terms). For which
of the following numbers q can three consecutive terms of the geometric sequence
be lengths of sides of a triangle?​

Answer 1

Given : The number q is a quotient of a geometric sequence (with positive terms)

To Find : For which of the following numbers q can three consecutive terms of the geometric sequence be lengths of sides of a triangle

A q= 0.25

B q=0.5

С q= 1.5

D) q= 2

Solution:

Sum of any two sides of a triangle is greater than third side.

Let say  one  side   is  16x

A q= 0.25

second side = 16x * 0.25 = 4x

third side = 4x * 0.25 = x

x + 4x  < 16x   Hence it can not form a triangle

A q= 0.5

second side = 16x * 0. 5 = 8x

third side = 8x * 0. 5 = 4x

8x + 4x  < 16x   Hence it can not form a triangle

q= 1.5

second side = 16x * 1. 5 = 24x

third side = 24x * 1. 5 = 36x

16 x + 24x  > 36 x

24x  + 36x  > 16x

16x + 36x  > 24x

Hence it can   form a triangle

q= 2

16x , 32x and 64x

16x + 32x < 64x   can not form a triangle

For  q= 1.5  three consecutive terms of the geometric sequence

be lengths of sides of a triangle

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