Let a₁, a₂, a₃, be a geometric progression of positive real numbers. If _ i=5 ⁸ a_i _ i=1 ⁴ a_i = 81 and a₃ a₆ = 19683 , then the value of a₂ + a₄ is equal to

Step-by-step solution

Let the first term of the geometric progression be a and the common ratio be r . Since all terms are positive, a > 0 and r > 0 . The ratio of the sums can be written as: a₅ + a₆ + a₇ + a₈ a₁ + a₂ + a₃ + a₄ = ar^4 + ar^5 + ar^6 + ar^7 a + ar + ar^2 + ar^3 Factoring out r^4 from the numerator: r^4(a + ar + ar^2 + ar^3) a + ar + ar^2 + ar^3 = r^4 We are given that this ratio is 81 , so: r^4 = 81 r = 3 (since r > 0 ). Next, we use the product condition: a₃ a₆ = (ar^2)(ar^5) = a^2 r^7 Substitute r = 3 into the equation: