Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

The first unit payment occurs at year three, and each later payment is seven years after the preceding one.

X=v3+v10+⋯+v7n−4X=v^3+v^{10}+\cdots+v^{7n-4}

Model

Model

Those discount factors form a finite geometric series with common ratio v to the seventh power.

X=v31−v7n1−v7X=v^3\frac{1-v^{7n}}{1-v^7}

Compute

Compute

Expressing one minus a power of v through annuity-immediate factors yields the stated ratio.

X=a7n+3‾∣−a3‾∣a7‾∣X=\frac{a_{\overline{7n+3}|}-a_{\overline3|}}{a_{\overline7|}}

Answer

Answer

That ratio is the expression printed in choice B.

a7n+3‾∣−a3‾∣a7‾∣(B)\boxed{\frac{a_{\overline{7n+3}|}-a_{\overline3|}}{a_{\overline7|}}\quad\text{(B)}}