When x is added to each of 8, 14, 20 and 30, then the numbers, so obtained,in this order form a proportion. What is the mean proportional between (x − 2) and (7x + 2)?

24

8, 14, 20, 30,

⇒ 8 + x, 14 + x, 20 + x, 30 + x

Using componendo and dividendo rule,

⇒ \(\frac{8 + x + 14 + x}{8 + x - 14 - x}\) = \(\frac{20 + x + 30 + x}{20 + x - 30 - x}\)

⇒ \(\frac{22 + x}{-6}\) = \(\frac{25 + x}{-10}\)

⇒ \(\frac{11 + x}{3}\) = \(\frac{25 + x}{5}\)

⇒ 55 + 5x = 75 + 3x

⇒ 2x = 20

⇒ x = 10

⇒ Mean Proportional = \(\sqrt {(x - 2)(7x + 2) }\)

⇒ \(\sqrt {(8)(72) }\) = \(\sqrt {8 × 8 × 9 }\)

⇒ A = 8 x 3 = 24.