Question

If b is the mean proportional between a and c, prove that (ab + bc) is the mean proportional between (a² + b²) and (b² + c²).

Answer 1

b is the mean proportional between a and c then

b² = ac            …(i)

Now if (ab + bc) is the mean proportional

between (a² + b²) and (b² + c²), then

(ab + bc)² = (a² + b²) (b² + c²)

Now L.H.S. = (ab + bc)²

= a²b + b²c² + 2ab²c

= a²(ac) + ac(c)² + 2a ac.c        ...[from (i)]

= ac(a² + c² + 2ac)

= ac(a + c)²

R.H.S. = (a² + b²)(b² + c²)

= (a² + ac)(ac + c²)             ...[from (i)]

= a(a + c) c(a + c)

= ac(a + c)²

∴ L.H.S. = R.H.S.