Question

Find four geometric means between 25 and 800.

Answer 1

Actually, you have to find four missed terms of geometrical sequence. Denote

$a_1=25,\\ a_6=800$

and r - ratio, then

$a_2=a_1\cdot r=25r,\\a_3=a_1\cdot r^2=25r^2,\\a_4=a_1\cdot r^3=25r^3,\\a_5=a_1\cdot r^4=25r^4,\\a_6=a_1\cdot r^5=25r^5$.

Since $a_6=800$, you have that $800=25r^5$, then

$r^5=\dfrac{800}{25} =32,\\r=2$.

Now substitude r=2 in each of unknown 4 terms:

$a_2=25\cdot 2=50,\\a_3=25\cdot 2^2=100,\\a_4=25\cdot 2^3=200,\\a_5=25\cdot 2^4=400$.

Answer: four geometric means between 25 and 800 are 50, 100, 200, 400.