Final answer:
The least positive integer n for which an < 0 is 22. The correct answer is A. 22.
Step-by-step explanation:
The question involves finding the least positive integer n for which an in a harmonic progression with a1=5 and a20=25 is less than 0. The series is in harmonic progression, which means the reciprocals of its terms are in arithmetic progression (AP).
If we denote the reciprocals of the terms by b1, b2, b3, ..., then b1 = 1/5 and b20 = 1/25. The difference d of this AP can be calculated using the formula d = (bn - b1)/(n-1). In this case, d = (1/25 - 1/5)/19 = -4/375.
The harmonic progression can be represented as:
a₁, a₂, a₃, ... = 5, 5 + 1/2, 5 + 1/3, ...
Given that a₁ = 5 and a₂₀ = 25, we can set up the equation:
5 + (1/20) * (a₂₀ - 5) = 5 + (1/20) * (25 - 5)
Simplifying the equation, we find:
a₂₀ = 5 + 20/20 = 25
So, the value of n for which aₙ < 0 is n = 22. Therefore, the answer is A. 22.