If `a ,b ,ca n dd` are in H.P., then prove that `(b+c+d)//a ,(c+d+a)//b ,(d+a+b)//c` and `(a+b+c)//d` , are in A.P.

Explanation

To prove that if \( a, b, c, d \) are in Harmonic Progression (H.P.), then the expressions \( \frac{b+c+d}{a}, \frac{c+d+a}{b}, \frac{d+a+b}{c}, \frac{a+b+c}{d} \) are in Arithmetic Progression (A.P.), we will follow these steps:

### Step 1: Understand the condition of H.P.
Since \( a, b, c, d \) are in H.P., their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d} \) are in Arithmetic Progression (A.P.). This means:
\[ \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} = \frac{1}{d} - \frac{1}{c} \]

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