Given a sequence: 19; 16; 13; 10; (a) Determine the common difference (c) Determine the 45th term. (b) Determine the nth term. (d) Which term of the sequence is -113?

The common difference is $\boxed{-3}$.

b) Determine the nth term.

Here, $a = 19$ and $d = -3$.

The nth term is $\boxed{22 - 3n}$.

c) Determine the 45th term.

The 45th term is $\boxed{-113}$.

d) Which term of the sequence is -113?

The 45th term is $\boxed{45}$.

2. $T_n = 9n - 4$ is the nth term of a linear number pattern.

a) Determine the first four terms of the sequence.

For $n=1$: $T_1 = 9(1) - 4 = 9 - 4 = 5$
For $n=2$: $T_2 = 9(2) - 4 = 18 - 4 = 14$
For $n=3$: $T_3 = 9(3) - 4 = 27 - 4 = 23$
For $n=4$: $T_4 = 9(4) - 4 = 36 - 4 = 32$
The first four terms are $\boxed{5, 14, 23, 32}$.

b) Which term is equal to 987?

Since $n$ must be an integer for a term number, $\boxed{987 \text{ is not a term in this sequence}}$.

3. Consider the pattern: $4 \times 7; 7 \times 15; 10 \times 23; 13 \times 31; ...$

a) Determine the nth term of the sequence.

$4, 7, 10, 13, ...$
This is an arithmetic sequence with first term $a_1 = 4$ and common difference $d_1 = 3$.
The nth term for the first factor is $A_n = 4 + (n-1)3 = 4 + 3n - 3 = 3n + 1$.

$7, 15, 23, 31, ...$
This is an arithmetic sequence with first term $a_2 = 7$ and common difference $d_2 = 8$.
The nth term for the second factor is $B_n = 7 + (n-1)8 = 7 + 8n - 8 = 8n - 1$.

The nth term of the sequence is $\boxed{24n^2 + 5n - 1}$.