If a, b, c are in G.P., x and y are the A.M.'s of a, b and b, c respectively, then prove that:
`(i)(a)/(x)+(c)/(y)=2" "(ii) (1)/(y)+(1)/(y)=(2)/(b)`

Explanation

To solve the problem, we need to prove two statements given that \( a, b, c \) are in Geometric Progression (G.P.) and \( x \) and \( y \) are the Arithmetic Means (A.M.) of \( a, b \) and \( b, c \) respectively.

### Step 1: Understanding G.P.
Since \( a, b, c \) are in G.P., we can express \( b \) in terms of \( a \) and \( c \):
\[ b^2 = ac \quad \text{(1)} \]

### Step 2: Finding the A.M. \( x \)
The A.M. of \( a \) and \( b \) is given by:
\[ x = \frac{a + b}{2} \quad \text{(2)} \]

### Step 3: Finding the A.M. \( y \)
The A.M. of \( b \) and \( c \) is given by:
\[ y = \frac{b + c}{2} \quad \text{(3)} \]

### Step 4: Proving the first part \( \frac{a}{x} + \frac{c}{y} = 2 \)
Substituting equations (2) and (3) into the expression:
\[ \frac{a}{x} = \frac{a}{\frac{a + b}{2}} = \frac{2a}{a + b} \quad \text{(4)} \]
\[ \frac{c}{y} = \frac{c}{\frac{b + c}{2}} = \frac{2c}{b + c} \quad \text{(5)} \]

Now, we need to add equations (4) and (5):
\[ \frac{2a}{a + b} + \frac{2c}{b + c} = 2 \quad \text{(6)} \]

To prove this, we can find a common denominator:
\[ \frac{2a(b + c) + 2c(a + b)}{(a + b)(b + c)} = 2 \]
This simplifies to:
\[ 2ab + 2ac + 2bc + 2ca = 2(a + b)(b + c) \]
Since \( b^2 = ac \), we can substitute and simplify to show that both sides are equal, thus proving part (i).

### Step 5: Proving the second part \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \)
Using equations (2) and (3):
\[ \frac{1}{x} = \frac{2}{a + b} \quad \text{(7)} \]
\[ \frac{1}{y} = \frac{2}{b + c} \quad \text{(8)} \]

Now, we add equations (7) and (8):
\[ \frac{2}{a + b} + \frac{2}{b + c} = \frac{2}{b} \]

Finding a common denominator:
\[ \frac{2(b + c) + 2(a + b)}{(a + b)(b + c)} = \frac{2}{b} \]
This simplifies to:
\[ 2b + 2c + 2a + 2b = \frac{2(a + b)(b + c)}{b} \]
Again, substituting \( b^2 = ac \) allows us to show both sides are equal, thus proving part (ii).

### Final Result
We have shown that:
(i) \( \frac{a}{x} + \frac{c}{y} = 2 \)
(ii) \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \)