<|begin_of_text|><|start_header_id|>system<|end_header_id|>

You are a helpful chatbot that always answers in Uzbek.<|eot_id|><|start_header_id|>user<|end_header_id|>

Menga n-chi uchburchak sonini topish uchun tenglama nima ekanligini va uni faqat o'rta maktab darajasidagi matematikadan foydalanib qanday isbotlash mumkinligini tushunish uchun oddiy tarzda tushuntiring. iltimos, har bir bosqichni LaTeX yordamida isbotlang.<|eot_id|><|start_header_id|>assistant<|end_header_id|>

Uchburchakli son - bu 1 dan n gacha bo'lgan barcha butun sonlarning yig'indisi. Uchinchi uchburchakli sonni topish uchun tenglama quyidagi formula bilan berilgan: \begin{equation} T_n = \frac{n(n+1)}{2} \end{equation} Ushbu formulani isbotlash uchun biz matematik induksiya yordamida aniqlay olamiz: Asos bosqich: n = 1 uchun tenglama bizga T1 = 1 * (1 + 1) / 2 = 1 beradi, bu birinchi uchburchakli son. Induktiv bosqich: T_k = k * (k + 1) / 2. Endi biz formula n = n + 1, ya'ni T_k = k * (k + 1) uchun ham to'g'ri ekanligini isbotlashimiz kerak.<|eot_id|><|start_header_id|>user<|end_header_id|>

Matematikaviy induktsiyani batafsil tushuntirib bera olasizmi?<|eot_id|>