If A, G and Hare respectively the A.M., G.M. and H.M. of three positive numbers a, b and c, then the equation whose roots are a, b, c is

By definition, we haveA=a+b+c3⇒a+b+c=3AG=(abc)1/3⇒abc=G3and, 1H=1a+1b+1c33H=bc+ca+ababc3H=ab+bc+caG3⇒ab+bc+ca=3G3HThe equation whose roots are a, b and c isx3−S1x2+S2x−S3=0x3−(a+b+c)x2+(ab+bc+ca)x−abc=0x3−3Ax2+3G3Hx−G3=0  [Using (i), (ii) and (iii)]