Ralph is 27 years old and starting an IRA (individual retirement account). He is going to invest $200 at the beginning of each month. The account is expected to earn 2.65% interest, compounded monthly. How much money will Ricky have in his IRA when he retires, at age 65?

$157,419.08

$94,723.10

$13,183.51

$416,424.15

Question

Ralph is 27 years old and starting an IRA (individual retirement account). He is going to invest $200 at the beginning of each month. The account is expected to earn 2.65% interest, compounded monthly. How much money will Ricky have in his IRA when he retires, at age 65? 

 $157,419.08 

 $94,723.10 

 $13,183.51 

 $416,424.15

tkhunny · Answer

i = 0.0265 -- Annual Interest Rate
j = i/12 = 0.002208333... -- Monthly Interest Rate
r = 1+j -- Monthly Accumulation Rate

65 - 27 = 38
38*12 = 456

$200(r^{456} + r^{455} + r^{454} + ... + r^{2} + r)$

That's it.  Can you add it all up?

anonymous · Answer

i have to do from 456 all the way to 1?

tkhunny · Answer

Well, there are 456 payments.  You weren't expecting to get away with fewer, were you?

It's not that hard, or even discouraging, if you know enough about geometric series.  Can you add up a geometric series without burning up your calculator?

anonymous · Answer

i dont think so. i'll just figure it out.

tkhunny · Answer

You should know this:

$1 + r + r^{2} + ... + r^{n-1} = \dfrac{1 - r^{n}}{1-r}$

Similarly, 

$r + r^{2} + r^{3} + ... + r^{n} = \dfrac{r - r^{n+1}}{1-r}$