If P dollars are invested at the end of each year in an annuity that earns interest at an annual rate r, the amount in the account will be A dollars after n years, where $$n = \frac { \log \left[ \frac { A r } { P } + 1 \right] } { \log ( 1 + r ) } .$$ If $2,700 is invested each year in an annuity earning 15% annual interest, when will the account be
Worth $15,000?
A)10.0 years
B)5.6 years
C)2.4 years
D)-1.3 years
E)4.3 years

Question

Quizplus · Accepted Answer

To find when the account will be worth $15,000, we substitute $2,700 for P, 0.15 for r, and $15,000 for A in the given formula:$$n = \frac { \log \left[ \frac { 15000 \times 0.15 } { 2700 } + 1 \right] } { \log ( 1 + 0.15 ) }$$$$n = \frac { \log \left[ \frac { 2250 } { 2700 } + 1 \right] } { \log ( 1.15 ) }$$$$n = \frac { \log \left[ \frac { 5 } { 6 } + 1 \right] } { \log ( 1.15 ) }$$$$n = \frac { \log \left[ \frac { 11 } { 6 } \right] } { \log ( 1.15 ) }$$$$n = \frac { \log ( 1.8333 ) } { \log ( 1.15 ) }$$$$n \approx 4.3 \text{ years}$$Therefore, it will take approximately 4.3 years for the account to be worth $15,000.

If P Dollars Are Invested at the End of Each n=log⁡[ArP+1]log⁡(1+r).n = \frac { \log \left[ \frac { A r } { P } + 1 \right] } { \log ( 1 + r ) } .n=log(1+r)log[PAr​+1]​.

If P Dollars Are Invested at the End of Each $n = \frac { \log \left[ \frac { A r } { P } + 1 \right] } { \log ( 1 + r ) } .$