2026 TPSS P1 Q25

The first four terms in a sequence are given below.

\(\begin{aligned} T_1&=2^2+1=5 \\ T_2&=3^2+4=13 \\ T_3&=4^2+7=23 \\ T_4&=5^2+10=35 \end{aligned}\)

1. Find \(T_5\).[1]
2. Show that the \(n\)th term of the sequence, \(T_n\) is given by \(n^2+5n-1\).[2]
3. \(T_n\) and \(T_{n+1}\) are consecutive terms of the sequence. Find and simplify \(T_{n+1}-T_n\).[2]
4. Hence, explain why the difference of two consecutive terms of the sequence is even for all values of \(n\).[1]