How to Solve the Given Expression
The mathematical expression displayed needs to be simplified. Let’s examine it step by step:
Given:
[
1 - \frac{1}{1 - \frac{1}{a + 1}}
]
Step 1: Simplify the Inner Expression
Start with the innermost fraction:
[
1 - \frac{1}{a + 1}
]
To simplify this, find a common denominator:
The common denominator will be (a + 1), thus:
[
1 = \frac{a + 1}{a + 1}
]
Therefore:
[
1 - \frac{1}{a + 1} = \frac{a + 1}{a + 1} - \frac{1}{a + 1} = \frac{a + 1 - 1}{a + 1} = \frac{a}{a + 1}
]
Step 2: Substitute Back Into the Original Expression
Now substitute (\frac{a}{a + 1}) back into the original expression, replacing (1 - \frac{1}{a + 1}):
[
1 - \frac{1}{\frac{a}{a + 1}}
]
Step 3: Simplify the Outer Expression
Simplifying the above expression further, notice that:
[
\frac{1}{\frac{a}{a + 1}} = \frac{a + 1}{a}
]
Therefore, the expression becomes:
[
1 - \frac{a + 1}{a}
]
To simplify (1 - \frac{a + 1}{a}), use the common denominator (a):
[
1 = \frac{a}{a}
]
Thus:
[
1 - \frac{a + 1}{a} = \frac{a}{a} - \frac{a + 1}{a} = \frac{a - (a + 1)}{a} = \frac{a - a - 1}{a} = \frac{-1}{a}
]
The Correct Answer
From the simplification, the expression simplifies to:
[
\boxed{-\frac{1}{a}}
]
Therefore, the correct answer is (A) (-\frac{1}{a}).