\( A=a-\frac{1}{2-\frac{2}{2-\frac{2}{a}}} \quad M= \) Express \( A \) as a single fraction in its simplest form

To express \( A \) as a single fraction, let's start by simplifying the expression inside \( A \). Given: \[ A = a - \frac{1}{2 - \frac{2}{2 - \frac{2}{a}}} \] First, simplify the innermost fraction \( 2 - \frac{2}{a} \): \[ 2 - \frac{2}{a} = \frac{2a - 2}{a} = \frac{2(a - 1)}{a} \] Now substitute this back into the middle fraction: \[ 2 - \frac{2}{2 - \frac{2}{a}} = 2 - \frac{2}{\frac{2(a - 1)}{a}} = 2 - \frac{2a}{2(a - 1)} = 2 - \frac{a}{a - 1} \] To combine this, find a common denominator: \[ 2 - \frac{a}{a - 1} = \frac{2(a - 1) - a}{a - 1} = \frac{2a - 2 - a}{a - 1} = \frac{a - 2}{a - 1} \] Now substitute this back into the original equation for \( A \): \[ A = a - \frac{1}{\frac{a - 2}{a - 1}} = a - \frac{a - 1}{a - 2} \] Now, to combine these terms, we need a common denominator: \[ A = \frac{A(a - 2) - (a - 1)}{a - 2} = \frac{Aa - 2A - a + 1}{a - 2} \] Notice the mistake. Let's go back to: \[ A = a - \frac{a - 1}{a - 2} = \frac{a(a - 2) - (a - 1)}{a - 2} = \frac{a^2 - 2a - a + 1}{a - 2} = \frac{a^2 - 3a + 1}{a - 2} \] So, the final expression for \( A \) simplifies to: \[ A = \frac{a^2 - 3a + 1}{a - 2} \]