Given a sequence $\{a_n\}$ with $a_1 = \frac{1}{2}$ and $a_{n+1} = 1 - \frac{1}{a_n}$, find $a_5$.

1. Problem Description

Given a sequence $\{a_n\}$ with $a_1 = \frac{1}{2}$ and $a_{n+1} = 1 - \frac{1}{a_n}$, find $a_5$.

2. Solution Steps

We are given the recursive formula $a_{n+1} = 1 - \frac{1}{a_n}$.

We want to find $a_5$, so we need to find $a_2, a_3, a_4$ first.

$a_1 = \frac{1}{2}$

$a_2 = 1 - \frac{1}{a_1} = 1 - \frac{1}{\frac{1}{2}} = 1 - 2 = -1$

$a_3 = 1 - \frac{1}{a_2} = 1 - \frac{1}{-1} = 1 - (-1) = 1 + 1 = 2$

$a_4 = 1 - \frac{1}{a_3} = 1 - \frac{1}{2} = \frac{1}{2}$

$a_5 = 1 - \frac{1}{a_4} = 1 - \frac{1}{\frac{1}{2}} = 1 - 2 = -1$

3. Final Answer

$a_5 = -1$

The answer is C.