Question

Joyce’s teacher asks her to write the sequence formed by the number of red squares in each picture for the first five pictures. What sequence should Joyce write?

Answer 1

Joyce should write
`1,8,64,512,4096`

Explanation:

Let's analyze the procedure of how the pictures are formed.

Her teacher said the images were created by taking a red square, dividing it into 9 smaller squares, and removing the center square.

For the first two images we can see and count that there is 1 red square in the first picture and 8 red squares in the second picture.

For the third image, there are 8 squares formed by 8 red squares each one.

The number of red squares is 8 x 8 = 64

We can write the sequence as

1 , 8 , 64 ⇒ 1 , 8 ,
`8^(2)`

For the fourth picture we can do the same reasoning and figure out that there are 8 x 8 x 8 = 512 red squares in the fourth image.

The sequence so far is

1 , 8 , 64 , 512 ⇒ 1 , 8 ,
`8^(2)` ,
`8^(3)`

We can generalize the sequence and write that the sequence is :

`8^(n)` with
`n` ∈ IN ⇒ The first five numbers will be

`8^(0),8^(1),8^(2),8^(3),8^(4)` =
`1,8,64,512,4096`