Question

James folds a piece of paper in half several times, each time unfolding the paper to count how many equal parts he sees. After folding the paper about six times, it becomes too difficult to fold it again but he is curious how many parts the paper would be broken into if he could continue to fold it.

Answer 1

Answer:

Number of Equal parts will be governed by the equation

Uₙ=2ⁿ,

where n is the number of times he folds the paper

Step-by-step explanation:

Initially the Paper has 1 equal part

Let n=number of time paper was folded

If he folds it once(n=1), it has 2 equal parts.

If he folds it the again(n=2), it has 4 equal parts.

If he folds it the again(n=3), it has 8 equal parts.

This can be written as a Sequence

2,4,8...

Since the next term is gotten by doubling the previous part it is a geometric sequence.

Now, the first term, a =2

Common ratio, r=4/2=8/4=2

Since he wants to find out how many parts he will have after folding 6 times, we are required to find the 6th term.

The nth term of a geometric series

Uₙ=arⁿ⁻¹

n=6

U₆=2X2⁶⁻¹=2 X 2⁵=64

After folding the Paper six times, James is going to have 64 equal parts.

If he could continue to fold it, the number of parts will be determined by the number of times he folds it i.e. the nth term.

Uₙ=arⁿ⁻¹

Uₙ=2X2ⁿ⁻¹

=2ⁿ⁻¹⁺¹=2ⁿ