An annuity having n payments of 1 has a present value of X. The first payment is made at the end of three years and the remaining payments are made at seven-year intervals thereafter.

Determine X.

• [[math]]\frac{a_{\overline{7n+3}|} - a_{\overline{3}|}}{s_{\overline{3}|}}[[/math]]
• [[math]]\frac{a_{\overline{7n+3}|}-a_{\overline{3}|}}{a_{\overline{7}|}}[[/math]]
• [[math]]\frac{a_{\overline{7n+3}|}-a_{\overline{7}|}}{a_{\overline{3}|}}[[/math]]
• [[math]]\frac{a_{\overline{7n+3}|}-a_{\overline{7}|}}{a_{\overline{7}|}}[[/math]]
• [[math]]\frac{a_{\overline{7n+3}|}-a_{\overline{7}|}}{s_{\overline{3}|}}[[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: B

The present value is

[[math]] \begin{align*} v^3 + v^{10} + v^{17} + \cdots + v^{-4+7n}\\ = \frac{v^{3}-v^{3+7n}}{1-v^{7}}=\frac{\left(1-v^{3+7n}\right)-\left(1-v^{3}\right)}{1-v}=\frac{a_{\overline{3+7n}|}-a_{\overline{{{3}}}|}}{a_{\overline{{{7}}}|}}. \end{align*} [[/math]]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.