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Question

The lengths of the sides in a right triangle form three consecutive terms of a geometric sequence. Find the common ratio of the sequence. (There are two distinct answers. Enter your answers as a comma-separated list.)

Answer 1

 Let consecutive terms of a geometric sequence is

a, ar, and ar²

These are the lengths of the sides in a right triangle .

Using Pythagorean theorem:

(ar²)² =a² +(ar)²

a²r⁴ =a² +a²r²

r⁴ =1 +r²

r⁴- r²-1 =0

let r² =t

t² -t-1=0

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Solve the equation

 t =(1+sqrt (1+4))/2

    = (1+sqrt 5)/2

      =1.618

r² =1.618

r =1.27