Simple Annuities
$ a_{\overline{n}\lvert } = \frac{1-v^n}{i}$ $ a_{\overline{\infty}\lvert } = \frac{1}{i}$
$ \ddot{a}_{\overline{n}\lvert}= \frac{1-v^n}{d}$ $ \ddot{a}_{\overline{\infty}\lvert } = \frac{1}{d}$
$ \overline{a}_{\overline{n}\lvert}= \frac{1-v^n}{\delta}$ $ \overline{a}_{\overline{\infty}\lvert } = \frac{1}{\delta}$
The future value counterparts can be derived by multiplying the PV with $(1+i)^n$
Increasing and Decreasing Annuities
$PV_{o}= (P+\frac{Q}{i}) a_{\overline{n}\lvert } + (-\frac{nQ}{i})v^n$
$ (Ia)_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{i}$ $ (Ia)_{\overline{\infty}\lvert } =\frac{1+i}{i^2}$
$(I\ddot{a})_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{d}$ $(I\ddot{a})_{\overline{\infty}\lvert } =\frac{1}{d^2}$
$ (I\overline{a})_{\overline{n}\lvert } = \frac{\ddot{a}_{\overline{n}\lvert}-nv^n}{\delta}$ $ (I\overline{a})_{\overline{\infty}\lvert } =\frac{1}{\delta d}$
$ (\overline{I}\overline{a})_{\overline{n}\lvert } = \frac{\overline{a}_{\overline{n}\lvert}-nv^n}{\delta}$ $ (\overline{I}\overline{a})_{\overline{\infty}\lvert } =\frac{1}{\delta^2}$