Determine the perimeter of the 100th figure in the following pattern.?

Question

Assume that all sides are 1 cm long. Pattern is here: http://img227.imageshack.us/img227/1777/patternux8.png I got 398 cm but my teacher told me that wasn't it. Teach me, please. Thank you. All I need is the number of squares. I don't know the pattern. I'm fine with the perimeter.

Answer 1

We can solve this problem in two steps.
Step 1: Determine the number of squares in the 100th figure.
The series (in term of the number of squares) can be
a. 1, 2, 4, 8, 16, ... (assume that it is a geometric series, i.e. n term is 2^(n-1))
b. 1, 2, 4, 7, 11, ... (assuming that it is an arithmetic series, i.e. the series is 1, 1+1, 1+1+2, 1+1+2+3, 1+1+2+3+4), ...
A mathematician will tell you that in fact there are many many other series possible that starts with 1,2,4,...
So, if series a. is the case, then the number of squares is 2^99 which is a very big number and so I think this is not the answer your teacher wants.
If the series is b. then the number of squares is
1+1+2+3+4+...+99
=1+(1+2+3+4+...+99)
=1+(1+99)x99/2
=1+100x99/2
=1+50x99
=4951.
Step 2. Have fun finding the perimeter now that you know the number of squares.

Answer 2

The pattern that each figure doubles in length, while the formula for the perimeter is (2n+2) with n= number of squares... 2^0 is the first count of squares and 2^1 is the second so the 100th figure is 2^99. This means that n=(2^99) and therefore the perimeter should =(2[2^99])+2 or simplified (2^100)+2= ???

Answer 3

I assume the pattern is that every time you double the number of squares.

If that's the case then list make a list of how many squares are in the nth figure

1 1
2 2
3 4
4 8
5 16

from this pattern it is easily seen that in the nth figure there are 2^(n-1) squares. The perimeter is simply 2+2*number of squares
=2+2*2^(n-1)
=2+2^n
so for the100th figure
P=2+2^100 cm

Answer 4

The first three figures have 1, 2, and 4 squares.

Assume the nth figure is 2^(n-1) squares in succession.

the perimeter is twice the number of squares + 2

2 * 2 ^ (100-1) +2 = 2^100+2 cm

This is ~1,000,000,000,000,000, 000,000,000,000,000 cm.

Answer 5

The number of squares in the next one is the number of squares in the previous one times 2.

The 100th figure would have 2^99 squares

Answer 6

the area is f(x)=2x
so, there would be 200 squares
if each side is 1 cm long, then the perimeter would be 402cm

Answer 7

f(x) = 2^x

2^100 will give you the y coordinate, then just multiply that by two and add 2 for the ends to get the perimiter in units.