If `H_1, H_2, ,H_(20) are 20` harmonic means between 2 and 3, then `(H_1+2)/(H_1-2)+(H_(20)+3)/(H_(20)-3)=` a. 20 b.21 c. 40 d. 38

Options

A. 20
B. 21
C. 40
D. 38

Explanation

To solve the problem, we need to find the value of \((H_1 + 2)/(H_1 - 2) + (H_{20} + 3)/(H_{20} - 3)\) where \(H_1, H_2, \ldots, H_{20}\) are the 20 harmonic means between 2 and 3.

### Step 1: Understanding Harmonic Means
The harmonic means \(H_1, H_2, \ldots, H_{20}\) between two numbers \(a\) and \(b\) can be calculated using the formula for the \(n\) harmonic means between two numbers:
\[ H_k = \frac{2ab}{(n+1) \cdot a + (n-k) \cdot b + k \cdot a} \]
where \(k\) is the index of the harmonic mean.
...