Question

A personal account earmarked as a retirement supplement contains $242,200. Suppose $200,000 is used to establish an annuity that earns 8%, compounded quarterly, and pays $6000 at the end of each quarter. How long will it be until the account balance is $0? (Round your answer UP to the nearest quarter.)

Answer 1

Let after x quarters the account balance will be $0.

Then, by given condition,

200000*[1+(8/4)/100]^x = 6000*[1+{1+(8/4)/100}+{1+(8/4)/100}²+{1+(8/4)/100}³+.......+{1+(8/4)/100}^x-1]

i.e., 200000*(1.02)^x = 6000*[(1.02)^x-1]/[(1.02)-1]

i.e., 200000*(1.02)^x = 300000*[(1.02)^x-1]

i.e., -100000*(1.02)^x = -300000

i.e., (1.02)^x = 3

i.e., log(1.02)^x = log(3)

i.e., x*log(1.02) = log(3)

i.e., x = log(3)/log(1.02)

i.e., x 55

Therefore, after 55 quarters, the account balance will be $0.