If the 5th, 9th, and 16th terms of an arithmetic progression are also consecutive terms of a geometric progression, find the common ratio of the geometric progression.

Step 1: Apply the property of a Geometric Progression (GP). If A_5, A_9, and A_16 are consecutive terms of a GP, then the square of the middle term equals the product of the first and third terms. (A_9)^2 = A_5 × A_16 Substitute the expressions for A_5, A_9, and A_16: (a + 8d)^2 = (a + 4d)(a + 15d) Step 2: Expand both sides of the equation. a^2 + 2(a)(8d) + (8d)^2 = a(a + 15d) + 4d(a + 15d) a^2 + 16ad + 64d^2 = a^2 + 15ad + 4ad + 60d^2 a^2 + 16ad + 64d^2 = a^2 + 19ad + 60d^2 Step 3: Simplify the equation to find the relationship between a and d. Subtract a^2 from both sides: 16ad + 64d^2 = 19ad + 60d^2 Rearrange the terms to one side: 64d^2 - 60d^2 = 19ad - 16ad 4d^2 = 3ad Since d cannot be zero (otherwise all terms of the AP would be the same, and thus the GP terms would also be the same, leading to a common ratio of 1, which is a trivial case and usually not what's intended in such problems), we can divide both sides by d: 4d = 3a Express a in terms of d: a = (4d)/(3) Step 4: Calculate the common ratio of the GP. The common ratio r of the GP is given by (A_9)/(A_5). r = (a + 8d)/(a + 4d) Substitute a = (4d)/(3) into the expression for r: r = (4d)/(3) + 8d(4d)/(3) + 4d To simplify, find a common denominator for the terms in the numerator and denominator: r = (4d)/(3) + (24d)/(3)(4d)/(3) + (12d)/(3) r = (4d + 24d)/(3)(4d + 12d)/(3) r = (28d)/(3)(16d)/(3) Multiply by the reciprocal of the denominator: r = (28d)/(3) × (3)/(16d) Cancel out 3 and d: r = (28)/(16) Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4: r = (28 ÷ 4)/(16 ÷ 4) r = (7)/(4) The common ratio of the GP is (7)/(4). Send me the next one 📸