3 A sequence is defined by the recurrence relation \(5 t _ { n + 1 } - 4 t _ { n } = 3 n ^ { 2 } + 28 n + 6\), for \(n \geqslant 0\), with \(t _ { 0 } = 7\).
- Find an expression for \(t _ { n }\) in terms of \(n\).
Another sequence is defined by \(\mathrm { v } _ { \mathrm { n } } = \frac { \mathrm { t } _ { \mathrm { n } } } { \mathrm { n } ^ { \mathrm { m } } }\), for \(n \geqslant 1\), where \(m\) is a constant.
- In each of the following cases determine \(\lim _ { n \rightarrow \infty } \mathrm {~V} _ { n }\), if it exists, or show that the sequence is divergent.
- \(m = 3\)
- \(m = 2\)
- \(m = 1\)