Question

If a,b,c,d are in continued proportion show that (ab+bc+cd²)=a²+b²+c²/b²+c²+d²

Answer 1

a, b, c, d in continued proportion

=> a/b = b/c = c/d

=> b²=ac, c²=bd, ad = bc ... we use these freely!

Then...

( a² - b² ) ( c² - d² )

= ( a² - ac ) ( bd - d² )

= ad ( a - c ) ( b - d )

= bc ( a - c ) ( b - d )

= ( ac - c² ) ( b² - bd )

= ( b² - c² ) ( b² - c² )

= ( b² - c² )²