Question

Visualize the following procedure: Tear in half a square piece of paper with an area of one square unit. Then tear it in half again. Predict the area of one of the resulting rectangles after 5 tears.

Answer 1

= 0.0625 in²

Explanation:

Area of a piece of paper = 1 unit²

then we tear this paper into half.

So area becomes 0.5 unit²

In this way the sequence formed will be

1, 0.5, 0.25, .........

This sequence is a geometric sequence

as
`(T_(2) )/(T_(1))= (0.5)/(1)=0.5`

common ratio r = 0.5

We have to find the 5th term of this sequence

Explicit formula of a geometric sequence is represented by

`T_(n) =a(r)^(n-1)`

Now
`T_(5) =1(0.5)^(5-1)`

= (0.5)⁴

= 0.0625 in²

After 5 tears area would be 0.0625 in²

Answer 2

`Area = (1)/(16)` square units

Explanation:

To predict the area of the piece of paper after 5 tears we can use a geometric sequence.

Each time a piece is torn, half of the previous area is lost. If the area of the first piece is 1 square unit then:

`a_1 = 1\\\\r = (1)/(2)\\\\a_2 = 1 *(1)/(2)\\\\a_2 = (1)/(2)\\\\a_n = 1((1)/(2))^(n-1)\\\\a_5 = 1((1)/(2))^(5-1)`

`a_5 = (1)/(16)` square units