Identify which of the following sequence is an A.P .
`1/2 , 2/3 , 3/4 , 4/5 , 5/6 …….`

Answer

To determine whether the sequence \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots \) is an arithmetic progression (A.P.), we need to check if the difference between consecutive terms is constant.

### Step 1: Identify the terms of the sequence
The given sequence is:
- First term \( a_1 = \frac{1}{2} \)
- Second term \( a_2 = \frac{2}{3} \)
- Third term \( a_3 = \frac{3}{4} \)

### Step 2: Calculate the difference between the first and second terms
To find the difference \( a_2 - a_1 \):
\[ a_2 - a_1 = \frac{2}{3} - \frac{1}{2} \]
To perform this subtraction, we need a common denominator. The least common multiple of 3 and 2 is 6.

Converting each fraction:
\[ \frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6} \]
Now, we can subtract:
\[ a_2 - a_1 = \frac{4}{6} - \frac{3}{6} = \frac{1}{6} \]

### Step 3: Calculate the difference between the second and third terms
Next, we calculate \( a_3 - a_2 \):
\[ a_3 - a_2 = \frac{3}{4} - \frac{2}{3} \]
Again, we need a common denominator. The least common multiple of 4 and 3 is 12.

Converting each fraction:
\[ \frac{3}{4} = \frac{9}{12}, \quad \frac{2}{3} = \frac{8}{12} \]
Now, we can subtract:
\[ a_3 - a_2 = \frac{9}{12} - \frac{8}{12} = \frac{1}{12} \]

### Step 4: Compare the differences
Now we compare the two differences we calculated:
- \( a_2 - a_1 = \frac{1}{6} \)
- \( a_3 - a_2 = \frac{1}{12} \)

Since \( \frac{1}{6} \neq \frac{1}{12} \), the differences are not equal.

### Conclusion
Since the differences between consecutive terms are not constant, we conclude that the sequence \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \ldots \) is **not an arithmetic progression (A.P.)**.

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