Good Day, I came across this question in a Form 6 Math textbook and it stomped me. I know it has to do with constructing the formula for compound interest and continuous interest but I think once I get help on question a) I can figure out b) and c) on my own. Thanks in advance.
You borrow $1 from a loan company. The company charges interest at the rate of 100% per annum.
After one year:
when this interest is added yearly, you owe 200% of $1 = $2,
when the interest is added half yearly, you owe 150% of 150% of $1 = $(1.5)^2 = $2.25
when the interest is added each qtr, you owe 125% of 125% of 125% of 125% of $1 = $(1.25)^4 = $2.44
Question: a) If the interest could be added continuously, have a guess at what you would owe after one year,
b) Work out what you would owe if the interest is added i) daily, ii) hourly, iii) by the second,
c) now repeat part a.
The general formula for the amount owed in dollars at the end of the year,
if interest is added $n$ times, is $ \left(1+\dfrac1n\right)^n.$
As $n$ approaches infinity, this approaches the base of natural logarithms, denoted $e$,
which is an irrational number, approximately $2.7182818$.