As large language models (LLMs) are adopted in an increasingly wide range of applications, user-model interactions have grown in both frequency and scale. Consequently, research has focused on evaluating the robustness of LLMs, an essential quality for real-world tasks. In this paper, we employ simple multi-turn follow-up prompts to evaluate models' answer changes, model accuracy dynamics across turns with Markov chains, and examine whether linear probes can predict these changes. Our results show significant vulnerabilities in LLM robustness: a simple "Think again" prompt led to an approximate 10% accuracy drop for Gemini 1.5 Flash over nine turns, while combining this prompt with a semantically equivalent reworded question caused a 7.5% drop for Claude 3.5 Haiku. Additionally, we find that model accuracy across turns can be effectively modeled using Markov chains, enabling the prediction of accuracy probabilities over time. This allows for estimation of the model's stationary (long-run) accuracy, which we find to be on average approximately 8% lower than its first-turn accuracy for Gemini 1.5 Flash. Our results from a model's hidden states also reveal evidence that linear probes can help predict future answer changes. Together, these results establish stationary accuracy as a principled robustness metric for interactive settings and expose the fragility of models under repeated questioning. Addressing this instability will be essential for deploying LLMs in high-stakes and interactive settings where consistent reasoning is as important as initial accuracy.