2023 SJI P1 Q7

The sum of the first \(n\) terms of a sequence \(\{T_n\}\), \(n\in\mathbb Z^+\), is given by \(S_n=n(n+1)\).

1. 1. Find an expression for the \(n\)th term, \(T_n\), of this sequence.
   2. Show that \(\{T_n\}\) is an arithmetic sequence.[5]

Another sequence \(\{U_n\}\), \(n\in\mathbb Z^+\), is defined as \(U_n=\mathrm e^{T_n}\).

2. 1. Show that \(\{U_n\}\) is a geometric sequence.
   2. Write down the common ratio of \(\{U_n\}\).[4]

A third sequence \(\{V_n\}\), \(n\in\mathbb Z^+\), is defined as \(V_n=\ln(T_n)\).

3. Show that \(\displaystyle\sum_{n=1}^N V_n=cN+\ln(N!)\), where \(c\) is a constant to be determined.[4]