Janelle used grey and white squares to make figures below that follow a pattern. The first 6 figures are shown. She recorded the number of grey and white squares.

1. Identify the pattern for the number of grey squares in even-numbered figures

Looking at the table for the number of grey squares:
Figure 1: 2
Figure 2: 2
Figure 3: 6
Figure 4: 6
Figure 5: 12
Figure 6: 12

For even figure numbers $n$, the number of grey squares is the same as the preceding odd figure $n-1$. Let's look at the sequence of grey squares for even figures $n = 2, 4, 6$:
$n=2$: $2 = 1 \times 2$
$n=4$: $6 = 2 \times 3$
$n=6$: $12 = 3 \times 4$

Let $k = \frac{n}{2}$. The number of grey squares for an even figure $n$ is $k \times (k+1)$.

2. Calculate the number of grey squares for Figure 52

For Figure 52, $n = 52$, which is an even number.
$k = \frac{52}{2} = 26$
Number of grey squares = $26 \times (26 + 1) = 26 \times 27 = 702$