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The volatile and non-linear nature of stock market data, particularly in the post-pandemic era, poses significant challenges for accurate financial forecasting. To address these challenges, this research develops advanced deep learning and machine learning algorithms to predict financial trends, quantify risks, and forecast stock prices, focusing on the technology sector. Our study seeks to answer the following question: "Which deep learning and supervised machine learning algorithms are the most accurate and efficient in predicting economic trends and stock market prices, and under what conditions do they perform best?" We focus on two advanced recurrent neural network (RNN) models, long short-term memory (LSTM) and Gated Recurrent Unit (GRU), to evaluate their efficiency in predicting technology industry stock prices. Additionally, we integrate statistical methods such as autoregressive integrated moving average (ARIMA) and Facebook Prophet and machine learning algorithms like Extreme Gradient Boosting (XGBoost) to enhance the robustness of our predictions. Unlike classical statistical algorithms, LSTM and GRU models can identify and retain important data sequences, enabling more accurate predictions. Our experimental results show that the GRU model outperforms the LSTM model in terms of prediction accuracy and training time across multiple metrics such as RMSE and MAE. This study offers crucial insights into the predictive capabilities of deep learning models and advanced machine learning techniques for financial forecasting, highlighting the potential of GRU and XGBoost for more accurate and efficient stock price prediction in the technology sector. [ABSTRACT FROM AUTHOR]