If the arithmetic means of two positive number a and b `(a gt b )` is twice their geometric mean, then find the ratio a: b

Options

A. one of the possible values of the first terms is `(2-sqrt(2))`
B. one of the possible vlaues of the first terms is `(2+sqrt(2))`
C. one of the possible values of the common ratio is `(sqrt(2)-1)`
D. one of the possible values of the common ratio is `(1)/(sqrt(2))`

Answer

A, B, D

Explanation

Let the first term be 'a' and common ratio will be 'r'.
According to the question, we have
a+ar=1 and `a/(a-r)=2`
Solving these, we get
`1/((1-r)(1+r))=2`
`rArr1-r^(2)=1/2`
`rArrr^(2)=1/2`
`r=pm1/sqrt2`
If `r=1/sqrt2,a=2-sqrt2` ltvbrgt If `r=-1/sqrt2,a=2-sqrt2`