Magnetic resonance imaging (MRI) is widely used to detect brain tumors, but its accuracy depends on the physician's expertise and often requires biopsy confirmation. Deep learning, especially in the field of computer vision, has revolutionized the diagnosis and classification of brain tumors using MRI. This study aims to design a sequential brain tumor detection and classification model based on deep learning and using fully convolutional neural networks. The proposed model consists of two steps: distinguishing non-neoplastic brain from neoplastic brain and determining the tumor type of the latter. Two models were trained using the Brain Tumor MRI Dataset. Four optimizers are studied for three classification tasks (Adam, Nesterov momentum, root-mean-square propagation, and adaptive gradient) to achieve the best results. Adam performed best at distinguishing tumor from non-tumor brains, with 100 % training accuracy and 98 % validation and test accuracy. Nesterov Momentum performed best at differentiating the three tumor types, with 100 % training accuracy and 92 % validation and testing accuracy. Nesterov also performed best on the third classification task, with 100 % training accuracy and 95 % validation and test accuracy. Nesterov-based sequence models show significant results compared to literature works. The proposed Nesterov momentum-based sequence model achieves high accuracy in MRI brain tumor detection and classification.