Consider the sequence `a _(n+1) = (1)/( 1 - a _(n)) , nge 1` Given that `a _(1) = (1)/(2)` find sum of first 100 terms of this sequence

Answer

To solve the problem, we need to find the sum of the first 100 terms of the sequence defined by the recurrence relation:

\[ a_{n+1} = \frac{1}{1 - a_n} \]
with the initial condition \( a_1 = \frac{1}{2} \).

### Step-by-Step Solution:

1. **Calculate the first few terms of the sequence:**
- Start with \( a_1 = \frac{1}{2} \).
- Calculate \( a_2 \):
\[ a_2 = \frac{1}{1 - a_1} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]
- Calculate \( a_3 \):
\[ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - 2} = \frac{1}{-1} = -1 \]
- Calculate \( a_4 \):
\[ a_4 = \frac{1}{1 - a_3} = \frac{1}{1 - (-1)} = \frac{1}{1 + 1} = \frac{1}{2} \]
- Calculate \( a_5 \):
\[ a_5 = \frac{1}{1 - a_4} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]
- Calculate \( a_6 \):
\[ a_6 = \frac{1}{1 - a_5} = \frac{1}{1 - 2} = \frac{1}{-1} = -1 \]

2. **Identify the pattern:**
- The terms calculated are:
- \( a_1 = \frac{1}{2} \)
- \( a_2 = 2 \)
- \( a_3 = -1 \)
- \( a_4 = \frac{1}{2} \)
- \( a_5 = 2 \)
- \( a_6 = -1 \)
- We can see that the sequence repeats every 3 terms:
- \( a_1, a_4, a_7, \ldots = \frac{1}{2} \)
- \( a_2, a_5, a_8, \ldots = 2 \)
- \( a_3, a_6, a_9, \ldots = -1 \)

3. **Count the occurrences of each term in the first 100 terms:**
- Since the sequence repeats every 3 terms, we can find how many complete cycles fit into 100 terms:
- There are \( \frac{100}{3} = 33 \) complete cycles with a remainder of 1.
- Thus, in the first 100 terms:
- \( \frac{1}{2} \) appears \( 34 \) times (33 from complete cycles + 1 from the first term).
- \( 2 \) appears \( 33 \) times.
- \( -1 \) appears \( 33 \) times.

4. **Calculate the sum of the first 100 terms:**
- The sum can be calculated as follows:
\[ \text{Sum} = \left(34 \times \frac{1}{2}\right) + \left(33 \times 2\right) + \left(33 \times -1\right) \]
- Calculate each part:
- \( 34 \times \frac{1}{2} = 17 \)
- \( 33 \times 2 = 66 \)
- \( 33 \times -1 = -33 \)
- Now sum these values:
\[ \text{Sum} = 17 + 66 - 33 = 50 \]

### Final Answer:
The sum of the first 100 terms of the sequence is \( \boxed{50} \).