Question

119. Beatrice decides to deposit $100 per month at the end of every month in a bank with an annual interest rate of 5.5% compounded monthly.

a. Write a geometric series to show how much she will accumulate in her account after one year.

Answer 1

Beatrice will accumulate $1230.72 at the end of the year.

Explanation:

We can write:

`(1+(0.055)/(12))=z`

`100=C` for deposits

The first month would have only the deposit reflected in her balance, then, expanding some steps of the calculation would yield:

`S_(1)=C\\S_(2)=Cz + C\\S_(3)=Cz^(2)+Cz + C\\S_(n)=Cz^(n-1) +...+Cz+C`

A geometric series is given by:

`1+x+...+x^(m)=(x^(m+1)-1)/(x-1)`

Translating our series to the short form:

`S_(n)=C((z^(n)-1)/(z-1) )`

plugin in the values for the 12 month gives:

`S_(12)=100(((1+(0.055)/(12))^(12) -1)/((0.055)/(12)))=1230.7169`