Geometric sequence, find the best interest option over a year

[thekopite]

Homework Statement

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.

Homework Equations

The Attempt at a Solution

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.

 

[eumyang]

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...

Actually, the amount doesn't increase to infinity. Try putting in larger and larger values of x into a graphing calculator or spreadsheet and see what happens.

 

[Curious3141]

Homework Statement

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
Investig!te to find the maximum possible amount in the man's account after one year.

Homework Equations

The Attempt at a Solution

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 akgeometric 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.

Hmm...sounds like they want you to discover the letter 'e'.

Look up Napier's constant, also known as the base of natural logarithms.