If `a_(1),a_(2),a_(3),"......"` be in harmonic progression with `a_(1)=5` and `a_(20)=25`. The least positive integer n for which `a_(n)lt0` is

Options

A. 22
B. 23
C. 24
D. 25

Explanation

To solve the problem, we need to find the least positive integer \( n \) for which \( a_n < 0 \) given that the sequence \( a_1, a_2, a_3, \ldots \) is in harmonic progression (HP) with \( a_1 = 5 \) and \( a_{20} = 25 \).

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
If a sequence is in harmonic progression, then the reciprocals of the terms are in arithmetic progression (AP). Therefore, we can denote:
\[ b_n = \frac{1}{a_n} \]
where \( b_n \) is in AP.

2. **Finding the First and Twentieth Terms**:
Given:
\[ a_1 = 5 \quad \Rightarrow \quad b_1 = \frac{1}{5} \]
\[ a_{20} = 25 \quad \Rightarrow \quad b_{20} = \frac{1}{25} \]

3. **Finding the Common Difference**:
The general term of an AP can be expressed as:
\[ b_n = b_1 + (n-1)d \]
For \( n = 20 \):
\[ b_{20} = b_1 + 19d \]
Substituting the known values:
\[ \frac{1}{25} = \frac{1}{5} + 19d \]
Rearranging gives:
\[ 19d = \frac{1}{25} - \frac{1}{5} \]
Finding a common denominator (which is 25):
\[ \frac{1}{5} = \frac{5}{25} \quad \Rightarrow \quad 19d = \frac{1 - 5}{25} = -\frac{4}{25} \]
Therefore:
\[ d = -\frac{4}{25 \times 19} = -\frac{4}{475} \]

4. **Finding the General Term**:
The general term \( b_n \) can now be written as:
\[ b_n = \frac{1}{5} + (n-1) \left(-\frac{4}{475}\right) \]
Simplifying this:
\[ b_n = \frac{1}{5} - \frac{4(n-1)}{475} \]
Converting \( \frac{1}{5} \) to have a common denominator of 475:
\[ b_n = \frac{95}{475} - \frac{4(n-1)}{475} = \frac{95 - 4(n-1)}{475} \]
Thus:
\[ b_n = \frac{99 - 4n}{475} \]

5. **Finding When \( a_n < 0 \)**:
We need \( a_n < 0 \) which implies \( b_n = \frac{1}{a_n} < 0 \). This occurs when:
\[ 99 - 4n < 0 \]
Solving for \( n \):
\[ 99 < 4n \quad \Rightarrow \quad n > \frac{99}{4} = 24.75 \]
The least positive integer \( n \) that satisfies this condition is:
\[ n = 25 \]

### Final Answer:
The least positive integer \( n \) for which \( a_n < 0 \) is \( \boxed{25} \).