Pembahasan Soal Ujian Profesi Aktuaris
| Institusi | : | Persatuan Aktuaris Indonesia (PAI) |
| Mata Ujian | : | Matematika Keuangan |
| Periode Ujian | : | April 2019 |
| Nomor Soal | : | 21 |
SOAL
Dari persamaan-persamaan berikut, yang manakah yang benar?
- \({\ddot a_{\left. {\overline {\, {n + 1} \,}}\! \right| }} = 1 + {a_{\left. {\overline {\, n \,}}\! \right| }}\)
- \({S_{\left. {\overline {\, n \,}}\! \right| }} = {(1 + i)^n}{a_{\left. {\overline {\, n \,}}\! \right| }}\)
- \(\frac{1}{d} = \sum\limits_{k = 1}^\infty {{v^{k – 1}}} \)
- Hanya 1
- 1 dan 2
- 1 dan 3
- 2 dan 3
- 1, 2, dan 3
| Diketahui | - \({\ddot a_{\left. {\overline {\, {n + 1} \,}}\! \right| }} = 1 + {a_{\left. {\overline {\, n \,}}\! \right| }}\)
- \({S_{\left. {\overline {\, n \,}}\! \right| }} = {(1 + i)^n}{a_{\left. {\overline {\, n \,}}\! \right| }}\)
- \(\frac{1}{d} = \sum\limits_{k = 1}^\infty {{v^{k – 1}}} \)
|
| Rumus yang digunakan | \({a_{\left. {\overline {\, n \,}}\! \right| }} = v + {v^2} + {v^3} + … + {v^n}\) |
| Proses pengerjaan | Pernyataan (1) benar, sebab :
\(1 + {a_{\left. {\overline {\, n \,}}\! \right| }} = 1 + \left\{ {v + {v^2} + {v^3} + … + {v^n}} \right\} = {\ddot a_{\left. {\overline {\, {n + 1} \,}}\! \right| }}\)Pernyataan (2) benar, sebab :
\({(1 + i)^n}{a_{\left. {\overline {\, n \,}}\! \right| }} = {(1 + i)^n}\left( {\frac{{1 – {{(1 + i)}^{ – n}}}}{i}} \right) = \left( {\frac{{{{(1 + i)}^n} – 1}}{i}} \right) = {S_{\left. {\overline {\, n \,}}\! \right| }}\) Pernyataan (3) juga benar, sebab :
\(\sum\limits_{k = 1}^\infty {{v^{k – 1}}} = 1 + v + {v^2} + {v^3} + .{\rm{ }}.{\rm{ }}. = \frac{1}{{1 – v}} = \frac{1}{{1 – \frac{1}{{1 + i}}}} = \frac{1}{d}\) |
| Jawaban | e. 1, 2, dan 3 |
