Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

Express both values through the same n-year accumulation excess, one plus i to n minus one.

X=21.8sn‾∣ 0.109=200(1.109n−1)X=21.8s_{\overline n|\,0.109}=200(1.109^n-1)

Model

Model

The finite annuity accumulation is proportional to that excess, while the spaced perpetuity value is inversely proportional to it.

X=192081.109n−1X=\frac{19208}{1.109^n-1}

Compute

Compute

Equating the two produces a square equation with positive root 9.8.

(1.109n−1)2=96.04,1.109n−1=9.8(1.109^n-1)^2=96.04,\qquad 1.109^n-1=9.8

Answer

Answer

Substitution into either value expression gives 1,960, which is choice C.

X=200(9.8)=1960(C)\boxed{X=200(9.8)=1960\quad\text{(C)}}