Sally is making square patterns with her square tiles using the pattern shown. She has black and white tiles. Find the size of her new square,

1. Relate the number of tiles added to the odd-number sequence.

The tiles added are
$3,5,7,\ldots$
This is an arithmetic sequence with first term $3$ and common difference $2$.

Let $65$ be the $n$th term of this sequence:
$3+2(n-1)=65$

2. Find the resulting square size.

$3+2(n-1)=65$
$2(n-1)=62$
$n=32$

Adding $3$ tiles changes Pattern 1 to Pattern 2, adding $5$ tiles changes Pattern 2 to Pattern 3, and so on.

So the $32$nd addition gives Pattern $33$.

Hence the new square is
$33 \times 33.$

1. Form an equation connecting the number of tiles added to the size of the new square.

For a new square of size $n\times n$, the number of tiles just added is
$2n-1.$
Given that she has just added $4k+3$ tiles,
$2n-1=4k+3.$

2. Solve for the new square size in terms of $k$.

$2n-1=4k+3$
$2n=4k+4$
$n=2k+2$
Hence the size of the new square is
$(2k+2)\times(2k+2).$