In the following ordinary annuity the interest is compounded

In the following ordinary annuity, the interest is compounded with each payment, and the payment is made at the end of the compounding period. Find the amount of time needed for the sinking fund to reach the given accumulated amount. (Round your answer to two decimal places.) $4500 yearly at 6% to accumulate $100,000. yr

Solution

Interest rate = 6% p.a.

Compounding yearly

yearly paid amount = 4500

Final amount = 100000

No of years?

4500 deposited at the beginning of every year at the end of 30 years amounted to 100000

Equivalent principal = I 4500 for n years II 4500 for n-1 years, .....

Hence 4500(1+0.6)^k = 100000 where k = n(n+1)/2

Or 1.06^k =22.22

k = ln 22.22/ln 1.06 = 3.100/0.058 =51.65

When 53.20 = n(n+1)/2

n^2+n-106.40 = 0

n = 9.827 years

Approximately 9 years and 9 months