Sequences

Arithmetic with $a = 3$ and $d = 4$, so $u_n = 3 + 4(n-1) = 4n-1$ and $u_{20} = 79$.

Geometric with $a = 2$ and $r = 3$: $u_5 = 2 \times 3^4 = 2 \times 81 = 162$.

$u_2 = 3(5) - 4 = 11$, then $u_3 = 3(11) - 4 = 29$.

The terms halve each time and shrink towards zero. A geometric sequence converges to 0 whenever $|r| < 1$.

The terms never settle towards a single value, so no limit exists. Note that it IS bounded, which shows bounded does not imply convergent.

Divide top and bottom by $n^2$: $\dfrac{5 - \frac1n}{2 + \frac{3}{n^2}} \to \dfrac52$. Equal degrees give the ratio of leading coefficients.