Question

You need to accumulate $10,000. To do so, you plan to make deposits of $1,100 per year, with the first payment being made a year from today, in a bank account that pays 7 percent annual interest. Your last deposit will be less than $1,100 if less is needed to round out to $10,000 or more than $1,100 if more is needed to round out to $10,000. How many years will it take you to reach your $10,000 goal? How large will the last deposit be? Show your work below

Answer 1

Future value of anuity (i.e annual payments ) = (c/i) x [ (1+i)ⁿ - 1 ]

where

c = annual payments = 1100

i = interest rate = 7%

n = no of years

10000 = (1100/0.07) x [ 1.07ⁿ - 1]  

10000*0.07/1100 = 1.07ⁿ -1

0.636 + 1 = 1.07ⁿ

1.636 = 1.07ⁿ

1.07^7.27. = 1.07ⁿ

n = 7.27 years

Since the last payment can be made in a greater amount to reach 10000 , an excess payment will be done in the end of 7th year.

using Future value of anuity for 6 years

FV = (1100/0.07) x [ 1.07⁶ - 1]  

FV = 7868.62 at the end of 6 years

Now this value will grow by 7% till the end of 7 years

So 7868.62 x 1.07 = 8419.42

So an excess of 1580.58 (i.e 10000 - 8419.42) needs to be paid to reach 10,000.

Thus final payment = 1580.58

Thanks if you have any doubts do leave a comment and let me know