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 find the sum of the first 100 terms of the sequence defined by \( a_{n+1} = \frac{1}{1 - a_n} \) with \( a_1 = \frac{1}{2} \), we can follow these steps:

### Step 1: Calculate the first few terms of the sequence.

1. **Calculate \( a_2 \)**:
\[ a_2 = \frac{1}{1 - a_1} = \frac{1}{1 - \frac{1}{2}} = \frac{1}{\frac{1}{2}} = 2 \]

2. **Calculate \( a_3 \)**:
\[ a_3 = \frac{1}{1 - a_2} = \frac{1}{1 - 2} = \frac{1}{-1} = -1 \]

3. **Calculate \( a_4 \)**:
\[ a_4 = \frac{1}{1 - a_3} = \frac{1}{1 - (-1)} = \frac{1}{2} \]

4. **Calculate \( a_5 \)**:
\[ a_5 = \frac{1}{1 - a_4} = \frac{1}{1 - \frac{1}{2}} = 2 \]

5. **Calculate \( a_6 \)**:
\[ a_6 = \frac{1}{1 - a_5} = \frac{1}{1 - 2} = -1 \]

### Step 2: Identify the pattern in the sequence.

From the calculations, we observe the sequence:
- \( a_1 = \frac{1}{2} \)
- \( a_2 = 2 \)
- \( a_3 = -1 \)
- \( a_4 = \frac{1}{2} \)
- \( a_5 = 2 \)
- \( a_6 = -1 \)

The sequence repeats every three terms: \( \frac{1}{2}, 2, -1 \).

### Step 3: Determine the number of complete cycles in the first 100 terms.

Since the sequence repeats every 3 terms, we can find how many complete cycles fit into 100 terms:
\[ \text{Number of complete cycles} = \frac{100}{3} = 33 \text{ complete cycles} \text{ and } 1 \text{ extra term.} \]

### Step 4: Calculate the sum of one complete cycle.

The sum of one complete cycle is:
\[ \text{Sum of one cycle} = \frac{1}{2} + 2 - 1 = \frac{1}{2} + 1 = \frac{3}{2} \]

### Step 5: Calculate the total sum for 100 terms.

1. **Sum from complete cycles**:
\[ \text{Sum from 33 cycles} = 33 \times \frac{3}{2} = \frac{99}{2} = 49.5 \]

2. **Add the extra term** (which is \( a_1 = \frac{1}{2} \)):
\[ \text{Total sum} = 49.5 + \frac{1}{2} = 49.5 + 0.5 = 50 \]

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

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