Write the next term of the sequence: `1/6,1/3,1/2`……..

Answer

To find the next term of the sequence \( \frac{1}{6}, \frac{1}{3}, \frac{1}{2} \), we can follow these steps:

### Step 1: Identify the terms of the sequence
The given terms are:
- First term \( a_1 = \frac{1}{6} \)
- Second term \( a_2 = \frac{1}{3} \)
- Third term \( a_3 = \frac{1}{2} \)

### Step 2: Check for a pattern
We need to determine if the sequence is an arithmetic progression (AP) or a geometric progression (GP).

#### Check for Arithmetic Progression:
To check if it is an arithmetic progression, we calculate the differences between consecutive terms.

- Difference between the second and first term:
\[ a_2 - a_1 = \frac{1}{3} - \frac{1}{6} \]
To perform this subtraction, we find a common denominator:
\[ = \frac{2}{6} - \frac{1}{6} = \frac{1}{6} \]

- Difference between the third and second term:
\[ a_3 - a_2 = \frac{1}{2} - \frac{1}{3} \]
Again, we find a common denominator:
\[ = \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]

Since both differences are equal (\( \frac{1}{6} \)), the sequence is an arithmetic progression with a common difference \( d = \frac{1}{6} \).

### Step 3: Find the next term
To find the next term \( a_4 \), we add the common difference \( d \) to the last term \( a_3 \):
\[ a_4 = a_3 + d = \frac{1}{2} + \frac{1}{6} \]

### Step 4: Perform the addition
To add \( \frac{1}{2} \) and \( \frac{1}{6} \), we need a common denominator:
- The least common multiple of 2 and 6 is 6.

Convert \( \frac{1}{2} \) to sixths:
\[ \frac{1}{2} = \frac{3}{6} \]

Now we can add:
\[ a_4 = \frac{3}{6} + \frac{1}{6} = \frac{4}{6} \]

### Step 5: Simplify the result
Now, simplify \( \frac{4}{6} \):
\[ \frac{4}{6} = \frac{2}{3} \]

### Conclusion
Thus, the next term of the sequence is:
\[ \boxed{\frac{2}{3}} \]