# Thread: geometric sequence, find the best interest option over a year

1. ## geometric sequence, find the best interest option over a year

The Bank of Utopia offers an interest rate of 100% per annum with various options as to how the interest may be added. A man invests \$1000 and considers the following options.
Option A - Interest added annually at the end of the year.
Option B - Interest of 50% credited at the end of each half-year.
Option C, D, E, ... The Bank is willing to add interest as often as required, subject to (interest rate) x (number of credits per year) = 100
Investigate to find the maximum possible amount in the man's account after one year.

So I took 1000(1 + (1/x)^x as the amount in the man's account by the end of the year, where x is the number of credits per year. I'm fairly sure this amount increases to infinity as x increases, but the differences between the amounts as x (x remaining an integer) increases must tend toward zero (considering the question). Since this is a section on geometric series I'm wondering if I'm supposed to salvage a geometric series out of this and calculate it's sum to infinity, but I have no idea which series to look for. Any suggestions would be appreciated, cheers.

2. ## Re: geometric sequence, find the best interest option over a year

The sequence $(1 + 1/n)^n$ is increasing (see here and here) and tends to the Euler's number e (also known as the Napier's constant and the natural logarithm base). See in particular the section about compound interest.