Solution

If 2x+1, x+2, 2 and 5 are in proportion, then ⇒ 2x+1 : x+2 :: 2 : 5 ⇒ (2x + 1)/(x + 2) = 2/5 ⇒ 5 (2x + 1) = 2 (x + 2) ⇒ 10x + 5 = 2x + 4 ⇒ 10x – 2x = 4 – 5 ⇒ 8x = (-1) ⇒ x = -1/8 Let mean proportional between 3.5 (1 – x) and 8 (1 + x) be M, then 3.5 (1 – x) : M :: M : 8 (1 + x) ⇒ [3.5(1 + 1/8)]/M = M/[8 (1 – 1/8)] ⇒ M = 3.5 × 9/8 × 8 × 7/8 ⇒ M = (7 × 9 × 7)/(2 × 8) ⇒ M = (7 × 7 × 3 × 3)/(4 × 4) ⇒ M = √(7 × 7 × 3 × 3)/(4 × 4) ⇒ M = (7 × 3)/4 ⇒ M = 21/4 ∴ M = 5.25