My mom told me that when she grabbed her pen, her index finger becomes painful. afterward, the pain radiated to her wrist and to her lower arm. now, she has difficulty moving and using her hand as it is painful. should we go to the doctor?

Has she been having symptoms in those regions? It's hard to imagine that significant injury in healthy tissues occurred by simply grabbing a pen. Sprain of muscles, tendons or ligaments are possibilities here. In view of her discomfot I do recommend being examined by a physician. In the meantime apply cold compresses to painful or swollen areas. Ibuprofen could be taken.

Yes. Mother may have tendinitis of the finger. She may benefit from a steroid injection at the finger base and would also evaluate for carpal tunnel. F/u/w/ MD https://www.healthline.com/health/tendonitis-in-finger#outlook