a, b, c are the length of sides BC, CA, AB respectively of `DeltaABC` satisfying `log(1+(c )/(a))+log a-log b=log2`. a, b, c are in :
i) A.P.
ii) G.P.
iii) H.P.
iv) none

Options

A. A.P.
B. G.P.
C. H.P.
D. None

Answer

To solve the problem, we need to analyze the given equation involving the sides of triangle ABC, denoted as \( a \), \( b \), and \( c \). The equation provided is:

\[ \log\left(1 + \frac{c}{a}\right) + \log a - \log b = \log 2 \]

### Step 1: Simplify the logarithmic equation

Using the properties of logarithms, we can combine the logarithmic terms on the left side:

\[ \log\left(1 + \frac{c}{a}\right) + \log a - \log b = \log\left(\frac{a(1 + \frac{c}{a})}{b}\right) \]

This simplifies to:

\[ \log\left(\frac{a + c}{b}\right) = \log 2 \]

### Step 2: Exponentiate both sides

By exponentiating both sides of the equation, we eliminate the logarithm:

\[ \frac{a + c}{b} = 2 \]

### Step 3: Rearranging the equation

Multiplying both sides by \( b \):

\[ a + c = 2b \]

### Step 4: Analyze the relationship

The equation \( a + c = 2b \) indicates that the sum of the lengths of sides \( a \) and \( c \) is twice the length of side \( b \). This is a characteristic of an arithmetic progression (A.P.), where the middle term is the average of the two outer terms.

### Conclusion

Thus, we conclude that the sides \( a, b, c \) are in Arithmetic Progression (A.P.).

### Final Answer

The correct option is **i) A.P.**

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