Question

2 The half-life of a substance is defined as the period of time it takes for the amount of the

substance to decay by half. The sequence shows the amount of a substance that will remain
after a certain number of half-lives have elapsed.

1, 1/2, 1/4, 1/8

Describe this sequence.

Answer 1

The given sequence is

1, 1/2, 1/4, 1/8......

we can see that the ratio of the consecutive terms is the same. We have

1/2/1 = (1/4)/(1/2) = (1/8)/(1/4) = 1/2

This means that it is a geometric sequence

To find the amount of the substance left after 21 half lifes, we would find the 21st term of the sequence. The formula for finding the nth term of a geometric sequence is

an = a1r^(n - 1)

a1 = 1

n = 21

r = 0.5

Thus,

a21 = 1 * 0.5^(21 - 1) = 0.5^20

a21 = 0.00000095367

a21 = 1/1048576

The answer makes sense in the problem context because it follows the common ratio that was established at the begining.