# N (number of shape) 1 2 3 4 5 .... 12 A (area of shape) a2 3/4* a2 6/9 *a2 10/16 *a2 15/25 * a2 ..... ?????? The perimeter for these shapes stays the same, always...

N (number of shape)

1

2

3

4

5

....

12

A (area of shape)

a2

3/4* a2

6/9 *a2

10/16 *a2

15/25 * a2

.....

??????

The perimeter for these shapes stays the same, always being equal to 4a. However. the area is changing

The area for the first shape which is a square (where n=1) is a2.

a) Using fractions, give unsimplified expressions for the area of the other shapes and describe a pattern.

b) Use your pattern to determine both an unsimplified and then a simplified expression for the area of the shape where n=12

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Presumably by a2 you mean `a^2` , where a is the length of the sides of the square.

For simplicity we will assume that all of the shapes are rectangular prisms (all sides meet at right angles).

We know that the denominator is n^2. We need to find an expression for the numerator also in terms of n.

n 1 2 3 4 5

x_n 1 3 6 10 15

n^2 1 4 9 16 25

n^2/2 0.5 2 4.5 8 12.5

n/2 0.5 1 1.5 2 2.5

Therefore, the numerator is equal to:`1/2n^2+n/2`

Consequently, the unsimplified formula for area in terms of n is:

`A_n=(0.5n^2+0.5n)/n^2a^2`

For n=12:

`A_12=(0.5(12)^2+0.5(12))/12^2a^2=78/144a^2`

`A_12=13/24a^2`

N (number of shape) = 1, 2, 3, 4, 5, .... 12,

A (area of shape) = a2, 3/4* a2, 6/9 *a2, 10/16 *a2, 15/25 * a2, ..... ??????

The perimeter for these shapes stays the same, always being equal to 4a. That means the new shapes are being generated out of the same frame, just by distorting it.

Distortion of the square frame give rise to rhombus shapes of varying tilt, hence varying area.

Area is given by A= a^2 sinθ where, a is the legth of each side and θ is the angle of inclination (or tilt) of the rhombus.

Here, these angles work out to be 90, 48.6, 41.8, 38.7, and 36.9 degrees respectively.

Square being a special case of the rhombus, the angle of inclination being 90°.

Closer look at the pattern of area of different shapes reveals the denominators of the fractions are squares of consecutive integers, while numerators are in the sum of the numbers constituting an Arithmetic Progression.

Thus the table can be rewritten as: N (number of shape) = 1, 2, 3, 4, 5, .... 12,

A (area of shape) = ∑1/1^2 *a^2, ∑2/2^2 *a^2, ∑3/3^2 *a^2 ..... ∑12/12^2 *a^2 = ∑n/n^2 *a^2 = n(n+1)/2/n^2 *a^2.

Where ∑n denotes sum of the arithmetic series upto the nth natural number.

Therefore, the 12th term is ∑12/12^2 *a^2

= (12(12+1)/2)/144 *a^2

= 12(13/2)/144 *a^2

= 6(13)/144 *a^2

= 78/144 *a^2

= 13/24 *a^2.

These numbers should be in a table. N is in the top row and A is in the bottom row

this is how the these numbers were presented

N (number of shape)

1

2

3

4

5

....

12

A (area of shape)

a2

3/4* a2

6/9 *a2

10/16 *a2

15/25 * a2

.....

??????