Independent solution

How to solve this Geometrically Varying Annuities question

Setup

Setup

Let X be the first payment. Payment t is X times 1.02 to the power t minus 1 and is discounted t years.

Ct=X(1.02)t−1C_t=X(1.02)^{t-1}
v=(1.03)−1v=(1.03)^{-1}

Model

Model

The present value is a finite geometric series with common ratio 1.02 divided by 1.03.

200000=X∑t=1201.02t−11.03t200000=X\sum_{t=1}^{20}\frac{1.02^{t-1}}{1.03^t}
200000=Xv−(1.02)20v211−1.02v200000=X\frac{v-(1.02)^{20}v^{21}}{1-1.02v}

Compute

Compute

The series factor is 17.726695, giving X = 11,282.42. Grow that amount through 19 increases to obtain the last payment.

X=11282.42X=11282.42
C20=11282.42(1.02)19=16436.35C_{20}=11282.42(1.02)^{19}=16436.35

Answer

Answer

The final payment rounds to 16,436, so the answer is choice B.

C20≈16436(B)\boxed{C_{20}\approx16436\quad\text{(B)}}