If three positive numbers `a,b` and c are in AP, GP and HP as well, than find their values.

Explanation

Since, `a,b,c` are in AP,GP and HP as well
`therefore " " b=(a+c)/(2) " " "…..(i)"`
` " " b^(2)=ac " " "…..(ii)"`
and `b= (2ac)/(a+c)" " "…..(iii)"`
Form Eq. (i) and (ii), we have
`((a+c))^(2)/(2)`
or `(a+c)^(2)=4ac`
or `(a+c)^(2)-4ac=0`
or `(a-c)^(2)=0`
`therefore " "a=c " " "........(iv)"`
On putting `c=a` in Eq. (i), we get `b=(a+a)/(2)=a " " "......(v)"`
From Eqs. (iv) and (v), `a=b=c`, thus the three numbes will be equal.