Question

An illustration shows three grids with boxes shaded in an X pattern inside. The first is a 6 by 6 grid labeled as term 1, the second is an 8 by 8 grid labeled as term 2, and the third is a 10 by 10 grid labeled as term 3. In all the grids, the diagonal boxes extending from the corners to the center are shaded. In term 1, 12 boxes are shaded, in term 2, 16 boxes are shaded, and in term 3, 20 boxes are shaded.
What are the first three terms of the arithmetic sequence represented by the number of shaded tiles and the common difference?

(2 pts)
The first three terms are
,
, and
, and the common difference is

Answer 1

The number of shaded boxes in each grid corresponds to the first three terms of an arithmetic sequence. The terms provided are:

- Term 1 (6x6 grid): 12 shaded boxes
- Term 2 (8x8 grid): 16 shaded boxes
- Term 3 (10x10 grid): 20 shaded boxes

The first three terms of the arithmetic sequence are 12, 16, and 20.

To find the common difference \(d\), we can subtract the first term from the second term:

\[
d = 16 - 12 = 4
\]

We can also confirm the common difference by checking the difference between the second and third terms:

\[
d = 20 - 16 = 4
\]

Thus, the first three terms of the arithmetic sequence are 12, 16, and 20, and the common difference is 4.

In conclusion, the answer is:
The first three terms are **12**, **16**, and **20**, and the common difference is **4**.