If `a ,b ,c` are in A.P., prove that the following are also in A.P. (i) `1/(b c),1/(c a),1/(a b),` (ii) `b+c ,c+a ,a+b` (iii) `a(1/b+1/c),b(1/c+1/a),c(1/a+1/b)` (iv)`a^2(b+c),b^2(c+a),c^2(a+b)` (v)`1/(sqrt(b)+sqrt(c)),1/(sqrt(c)+sqrt(a)),1/(sqrt(a)+sqrt(b))`

Explanation

To prove that the given sequences are in Arithmetic Progression (A.P.), we will use the property that for three numbers \( x, y, z \) to be in A.P., the condition \( 2y = x + z \) must hold.

Given that \( a, b, c \) are in A.P., we can express this as:
\[ 2b = a + c \]

Now, let's prove each part step by step.

### (i) Prove that \( \frac{1}{bc}, \frac{1}{ca}, \frac{1}{ab} \) are in A.P.
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