Anais has access to 1001 bank accounts. The frst 1000 of them are client account

ID: 2960937 • Letter: A

Question

Anais has access to 1001 bank accounts. The frst 1000 of them are client accounts,

in which interest is calculated according to a nominal rate of 6%, convertible monthly, and

the last is a master account, in which interest is calculated according to a constant force

of interest of 0.09. At time zero, each client account starts out with a balance of $10, and

the master account starts empty. At the end of each month, she closes out one of the client

accounts and transfers the entire balance in it to the master account (so we have one less

account each month). What is the balance in the master account immediately following

the last transfer?

after first month the total amount in the master accountis

=(10*(1+0.06/12))

= (0.05)

after month 2 the total amount

=(0.05*(1+0.09/12)) +0.05

=(0.05*1.0075 +0.05)

=0.05*(1.0075+1)

after month 3 the total amount is

=(0.05*1.0075 +0.05) *1.0075 +0.05

=0.05 *(1.0075^2 +1.0075+1)

after month 4

=0.05*(1.0075^3+1.0075^2+1.0075+1)

continung like this

the last transaction will be at 1000th month

and the amount in the master account is

=0.05*(1.0075^999+1.0075^998 +........+1)

=0.05* 1*(1.0075^998 -1)/(1.0075-1)

=$11,540.45

the balance in the master account immediately following the last transfer =$11,540.45

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