If q is the mean proportional between p and r , then prove that p 3 + q 3 + r 3 p 2 q 2 r 2 = 1 p 3 + 1 q 3 + 1 r 3

Given, q is the mean proportional between p and r . Therefore, q 2 = p r . We have, p 3 + q 3 + r 3 p 2 q 2 r 2 = 1 p 3 + 1 q 3 + 1 r 3 Taking LHS L H S = p 3 + q 3 + r 3 p 2 q 2 r 2 = p 3 + q 2 q + r 3 p 2 × p r × r 2 = p 3 + p r × q + r 3 p 3 r 3 = p 3 p 3 r 3 + p r q p 3 r 3 + r 3 p 3 r 3 = 1 r 3 + q p r 2 + 1 p 3 = 1 r 3 + q q 2 2 + 1 p 3 = 1 r 3 + 1 q 3 + 1 p 3 = R H S Therefore L H S = R H S Hence, proved.

If q is the mean proportional between p and r , then prove that p 3 + q 3 + r 3 p 2 q 2 r 2 = 1 p 3 + 1 q 3 + 1 r 3

Given, q is the mean proportional between p and r . Therefore, q 2 = p r . We have, p 3 + q 3 + r 3 p 2 q 2 r 2 = 1 p 3 + 1 q 3 + 1 r 3 Taking LHS L H S = p 3 + q 3 + r 3 p 2 q 2 r 2 = p 3 + q 2 q + r 3 p 2 × p r × r 2 = p 3 + p r × q + r 3 p 3 r 3 = p 3 p 3 r 3 + p r q p 3 r 3 + r 3 p 3 r 3 = 1 r 3 + q p r 2 + 1 p 3 = 1 r 3 + q q 2 2 + 1 p 3 = 1 r 3 + 1 q 3 + 1 p 3 = R H S Therefore L H S = R H S Hence, proved.

Find the third proportional to 5 + 2 3 , 37 + 20 3

Given, Two numbers 5 + 2 3 and 37 + 20 3 Let the third proportional to 5 + 2 3 and 15 be x . If x , y and z are in continued proportion, then z is called third proportional. Therefore, 5 + 2 3 : 37 + 20 3 : : 37 + 20 3 : x ⇒ 5 + 2 3 37 + 20 3 = 37 + 20 3 x ⇒ x = 37 + 20 3 2 5 + 2 3 ⇒ x = 37 2 + 20 3 2 + 2 × 37 × 20 3 5 + 2 3 ⇒ x = 1369 + 1200 + 1480 3 5 + 2 3 Rationalize the denominator ⇒ x = 2569 + 1480 3 5 + 2 3 × 5 - 2 3 5 - 2 3 ⇒ x = 2569 + 1480 3 5 - 2 3 5 2 - 2 3 2 ⇒ x = 2569 5 - 2 3 + 1480 3 5 - 2 3 25 - 12 ⇒ x = 12845 - 5138 3 + 7400 3 - 8880 13 ⇒ x = 3965 + 2262 3 13 ⇒ x = 13 305 + 174 3 13 ⇒ x = 305 + 174 3 Hence, the required third proportional is 305 + 174 3 .

Find the third proportional to x y + y x and x y

Given, two numbers x y + y x and x y Let the third proportional to x y + y x and x y be z . If x , y and z are in continued proportion, then z is called third proportional. Therefore, x y + y x : x y : : x y : z ⇒ x y + y x x y = x y z ⇒ z = x y 2 x y + y x ⇒ z = x 2 y 2 x 2 + y 2 x y ⇒ z = x 2 y 2 × x y x 2 + y 2 ⇒ z = x 3 y x 2 + y 2 Hence, the required third proportional is x 3 y x 2 + y 2 .