Pembahasan Soal Ujian Profesi Aktuaris
SOAL
Tentukan kondisi yang harus dipenuhi oleh \(n,p,q,{\rm{ dan }}r\) sehingga persamaan berikut benar.
\({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\)
- \(p + q + r = n\)
- \(p + q = n{\rm{ dan }}n = r\)
- \(p + q = n{\rm{ dan }}p = r\)
- \(p + q = n{\rm{ dan }}q = r\)
- Tidak ada kondisi yang benar
| Diketahui | \({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\) |
| Rumus yang digunakan | \({a_{\left. {\overline {\, n \,}}\! \right| }} = \frac{{1 – {v^n}}}{i}\) \({S_{\left. {\overline {\, n \,}}\! \right| }} = \frac{{{{(1 + i)}^n} – 1}}{i} = \frac{{{v^{ – n}} – 1}}{i}\) \({v^t} = {(1 + i)^{ – t}}\) |
| Proses Pengerjaan | \({a_{\left. {\overline {\, n \,}}\! \right| }} = ({a_{\left. {\overline {\, p \,}}\! \right| }} + {S_{\left. {\overline {\, q \,}}\! \right| }}){v^r}\) \(\Leftrightarrow \frac{{1 – {v^n}}}{i} = (\frac{{1 – {v^p}}}{i} + \frac{{{v^{ – q}} – 1}}{i}){v^r}\) \(\Leftrightarrow \frac{{1 – {v^n}}}{i} = (\frac{{{v^{ – q}} – {v^p}}}{i}){v^r}\) \(\Leftrightarrow 1 – {v^n} = ({v^{ – q}} – {v^p}){v^r}\) \(\Leftrightarrow 1 – {v^n} = {v^{ – q + r}} – {v^{p + r}}\) \(\Leftrightarrow 1 – {v^n} = {v^{ – q + r}} – {v^{p + r}}{\rm{ (*)}}\) \({\rm{dari (*) diperoleh}}\) \({v^{ – q + r}} = 1\) \(\Leftrightarrow – q + r = 0\) \(\Leftrightarrow r = q\) \({\rm{selanjutnya }}\) \({v^n}{\rm{ = }}{v^{p + q}}\) \(\Leftrightarrow n = p + q\) |
| Jawaban | d. \(p + q = n{\rm{ dan }}q = r\) |