A bean stalk doubled its height every day. This was a magical bean stalk that grew from the magical beans that Jack planted, which he got from selling his mother's cow. The bean
stalk reached its full height in 100 days. How many days did it take to reach half its height?

The number of days it take to reach half its height is 99 days

Step-by-step explanation:

The given parameters are;

The height to which the beans stalk grew daily = Twice the initial height

The time duration for the bean stalk to reach maximum height = 100 days

Therefore, the growth of the bean stock is in the form of a geometric progression with a common ratio, r = 2

The formula for a  geometric progression = a×r⁽ⁿ⁻¹⁾

Therefore, the height of the bean is given by the formula a×r⁽ⁿ⁻¹⁾

Where;

The first term = a  = 1 (for analysis)

The number of terms (days) = n = 100 for 100 days

Therefore, we have;

After 100 days, the height of the bean = 1×2⁽¹⁰⁰⁻¹⁾ = 2⁹⁹ = 6.34 × 10²⁹

Half the height = 6.34 × 10²⁹/2 = 3.1291265 × 10²⁹

Which gives;

1×2⁽ⁿ⁻¹⁾ = 3.1291265 × 10²⁹

n - 1 = ㏒(3.1291265 × 10²⁹)/㏒(2) = 98

Which gives;

n - 1 = 98

n = 98 + 1 = 99 days.

1/800 is the right answer