If x is subtracted from each of the numbers 20, 37, 54 and 105, then the numbers so obtained in this order are in proportion. What is the mean proportional between (7x - 5) and (x + 1)?

As per question $$\dfrac {20-x}{37-x} = \dfrac{54-x}{105-x}$$

$$\Rightarrow 2100-20x-105x+x^2= 1998-37x-54x+x^2$$

$$\Rightarrow 125x-91x= 2100-1998$$

$$\Rightarrow 34x= 102 $$

$$\Rightarrow x =3 $$ 

mean proportional  between $$(7x-5) and (x+1) = \sqrt{(7x-5) (x+1)}$$

$$\Rightarrow \sqrt {(7\times 3-5)(3+1)}$$ put the value $$x$$

$$\Rightarrow \sqrt {16\times 4}$$

$$\Rightarrow \sqrt{64}$$

$$\Rightarrow 8 $$ Ans