Question

If a 1 (> 0), a 2, a 3, a 4, a 5 are in a G.P., a 2 + a 4 = 2 a 3 + 1 and 3 a 2 + a 3 = 2 a 4, then a 2 + a 4 + 2 a 5 is equal to _______.

Answer 1

The correct answer is 40
Let G.P. be a 1 = a , a 2 = ar , a 3 = ar 2, ……
$∵ 3a_2 + a_3 = 2a_4$
$⇒ 3ar + ar^2 = 2ar³$
⇒ 2ar² - r - 3 = 0
∴ r = -1 or $\frac{3}{2}$
∵ a1 = a > 0 then r ≠ -1
Now,

$a_2 + a_4 = 2a_3 + 1$
$ar + ar³ = 2ar² + 1$
$a ( \frac{3}{2} + \frac{27}{8} - \frac{9}{2} ) = 1$
$∴ a = \frac{8}{3}$
$∴ \frac{8}{3} ( \frac{3}{2} + \frac{27}{8} + \frac{81}{8} )$
= 40