Question

Let \( a_1, a_2, a_3, \dots \) be a harmonic progression with \( a_1 = 5 \) and \( a_{20} = 25 \). The least positive integer \( n \) for which \( a_n<0 \) is:

• A. 22
• B. 23
• C. 24
• D. 25

Hint:

When working with harmonic progressions, first convert them into arithmetic progressions by taking the reciprocals of the terms. This allows you to use the properties of arithmetic progressions to solve the problem.

Answer 1

The Correct Option is D

A harmonic progression (HP) is the sequence of numbers whose reciprocals form an arithmetic progression (AP). Let \( a_1, a_2, a_3, \dots \) be a harmonic progression. 
Then, the reciprocals \( \frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}, \dots \) form an arithmetic progression. Let the terms of this AP be denoted by \( b_1, b_2, b_3, \dots \), where: \[ b_n = \frac{1}{a_n} \] Given that \( a_1 = 5 \) and \( a_{20} = 25 \), we know the values of \( b_1 = \frac{1}{5} \) and \( b_{20} = \frac{1}{25} \). The general term of an arithmetic progression is given by: \[ b_n = b_1 + (n - 1) \cdot d \] where \( d \) is the common difference. Substituting the known values for \( b_1 \) and \( b_{20} \), we have: \[ \frac{1}{25} = \frac{1}{5} + (20 - 1) \cdot d \] 
Simplifying: \[ \frac{1}{25} = \frac{1}{5} + 19d \] \[ \frac{1}{25} - \frac{1}{5} = 19d \] \[ \frac{-4}{25} = 19d \] \[ d = \frac{-4}{25 \times 19} = \frac{-4}{475} \] Thus, the common difference is \( d = \frac{-4}{475} \). 
Now, the general term of the harmonic progression is: \[ a_n = \frac{1}{b_n} = \frac{1}{b_1 + (n - 1) \cdot d} \] Substituting \( b_1 = \frac{1}{5} \) and \( d = \frac{-4}{475} \), we get: \[ a_n = \frac{1}{\frac{1}{5} + (n - 1) \cdot \frac{-4}{475}} \] 
Simplifying the expression: \[ a_n = \frac{1}{\frac{1}{5} - \frac{4(n - 1)}{475}} \] Now, to find the least \( n \) such that \( a_n<0 \), we need to solve the inequality: \[ \frac{1}{5} - \frac{4(n - 1)}{475}<0 \] Multiplying through by 475: \[ 95 - 4(n - 1)<0 \] \[ 95 - 4n + 4<0 \] \[ 99 - 4n<0 \] \[ 4n>99 \] \[ n>\frac{99}{4} = 24.75 \] Thus, the least integer \( n \) is \( n = 25 \). Therefore, the correct answer is Option D.