Lesson 1

Consider a sequence \[T_{1}=1, T_{2}=3, T_{3}=7, T_{4}=15, T_{5}=31, T_{6}=63, \dotsc,\] defined recursively: \(T_{1}=1\) and \(T_{n}=2T_{n-1}+1\) for \(n>1\).

Problem. Find a closed-form formula for \(T_{n}\).