Question

figure#0 has 4 tiles figure#1 has 9 tiles figure#2 has 16 figure #3 has 25 tiles figure#4 has 36 tiles figure#5 has 49 tiles how many will figure #100 have,and what is the rule that will give the number of tiles in any figure in the pattern.

Answer 1

The figure number and the number of tiles are related by the number the formula

t = (f + 2)^2

The number of tiles (t) equals the figure number plus 2 the result of that squared.

For example for f =1

the number of tiles is t = (n + 2) ^2 = (1 + 2)^2 = 3^2 = 9

For f = 2

The number of tiles = (n+2)^2 = (2 + 2)^2 = 4^2 = 16

For f = 3

The number of tiles = (n + 2)^2 = (3 + 2)^2 =5^2 = 25

We can be fairly confident in the formula. So what about figure 100

t = (n + 2)^2 = 102^2 = 10404 tiles.

Your next problem is figure out what this will look like.

You are going to have to do a little reading as you draw this on a piece of paper.

Go back to figure 1. There is a square in the middle of the drawing. The square consists of 4 tiles. Draw them on a piece of paper. I can't reproduce them, but it looks like every drawing has a square with (n + 1)^2 tiles in it.

Look at figure 2. There is a square in the middle of the drawing. The figure number plus 1 all squared = the number of tiles in that square. t = (f + 1)^2. You should be able to see 9 squares in that central square.

Three will give you 16 squares in the middle square.

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Now there are arms coming off the square. Again each arm is related to the figure number.

In figure 1 the arm length is 2 tiles. It suggests 2*f is the number of tiles in both arms together.

Figure 2 has 3 tiles in each arm. That means that the total number of tiles used for 1 arm (f + 1) = (2 + 1) = 3

Figure 3 has 4 tiles in each arm. Total 8.

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Finally all figures have 1 tile sitting on top of everything else. No more than 1 and no less than 1. Just 1.

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Now we are ready to talk about figure 100.

The square in the middle was determined to be (f + 1)^2 = 101^2 = 10201

The arms were determined to be t = 2 *( f + 1) = 2 * 101 = . . . . .. . . . . .202

And there is one more square that every figure has . . . . . . . . . . . . . . . ..1

Total . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10404

Just what we got before. If you need more, leave a note about what you need, in detail.