Question

On the first day of each month, starting January 1, 1995, Smith deposits 100 in an acount earning i(12)=0.09, with interest credited the last day of each month. In addition,Smith deposits 1000 in the account every December 31. On what day does the account first exceed 100,000?

the answer show is need to be April 30,2013

Answer 1

We have 2 types of payment here of $100 at beginning of every month and $1000 at every year end. Therefore, time fo both will be different.

Let time for $100 annuity is m months and for $1000 annuity, time is n years.

Accumulated value = $100 x S__(0.0075,n) + $1000 x S__(.09,s) where s is year immediate n . and n is in months.

Interest is calculated monthly but deposit of $1000 is annually. So, following will be series

= 1000(1.0075)¹² + 1000(1.0075)^{12 x 2} +..............+ 1000(1.0075)^{12 x s}   

=1000 ((1.0075)¹² + (1.0075)²⁴ + ....................+(1.0075)^12s)

=1000 x (1.0075)¹² (1- (1.0075)^12s)/(1-(1.0075)¹²)

Solve the equation putting different values of n and s.

and s=18

n 18.3

Means, after 18 years and 4 months , accumulated value will exceed $100,000 first time.