Independent solution

How to solve this Annuities and Perpetuities question

Setup

Setup

Write payment t as the unknown first payment minus five for each elapsed payment interval.

10000=∑t=130X−5(t−1)1.05t10000=\sum_{t=1}^{30}\frac{X-5(t-1)}{1.05^t}

Model

Model

Discount the level portion and the arithmetic decrement on the same year-end schedule.

10000=Xa30‾∣ 0.05−5∑t=130(t−1)vt10000=Xa_{\overline{30}|\,0.05}-5\sum_{t=1}^{30}(t-1)v^t

Compute

Compute

The level-annuity factor is 15.37245 and the decrement adjustment is 843.11275.

10000=15.37245X−843.11275,X=705.3610000=15.37245X-843.11275,\qquad X=705.36

Answer

Answer

The first payment is approximately 705.36, so the listed answer is choice C.

X=705.36(C)\boxed{X=705.36\quad\text{(C)}}