The diagram shows dots and crosses in the first 3 patterns. (a) Complete the table. (b) Find (i) the number of crosses needed for pattern 45, (ii) the nth term for the sum of crosses and dots.

The number of dots for pattern 1 is 1 ($1^2$).
The number of dots for pattern 2 is 4 ($2^2$).
This suggests the number of dots for pattern $n$ is $n^2$.
For pattern 3, the number of dots is $3^2 = 9$.

The number of crosses for pattern 1 is 5.
The number of crosses for pattern 2 is 8.
The difference between consecutive terms is $8 - 5 = 3$. This is an arithmetic progression with a common difference of 3.
For pattern 3, the number of crosses is $8 + 3 = 11$.

For pattern 3, the number of dots is 9 and the number of crosses is 11.
The sum of crosses and dots is $9 + 11 = 20$.

The completed table is:

The sequence for the number of crosses is $5, 8, 11, \dots$.
This is an arithmetic progression with the first term $a = 5$ and the common difference $d = 8 - 5 = 3$.
The formula for the $n$th term of an arithmetic progression is $T_n = a + (n-1)d$.
Substitute $a=5$ and $d=3$:

Substitute $n=45$ into the formula:

The number of crosses needed for pattern 45 is $\boxed{\text{137}}$.

The sequence is $6, 12, 20, \dots$.
Calculate the first differences:
$12 - 6 = 6$
$20 - 12 = 8$
Calculate the second differences:
$8 - 6 = 2$
Since the second differences are constant, the $n$th term is a quadratic sequence of the form $T_n = an^2 + bn + c$.

The second difference is equal to $2a$:

The first term of the first differences is equal to $3a+b$:

Substitute $a=1$:

The first term of the sequence is equal to $a+b+c$:

Substitute $a=1$ and $b=3$:

Therefore, the $n$th term for the sum of crosses and dots is $T_n = 1n^2 + 3n + 2$.
The $n$th term for the sum of crosses and dots is $\boxed{n^2 + 3n + 2}$.