a)
\( q = \left( 1 + \frac{i}{m} \right) - 1 = \left( 1 + \frac{0.04}{4} \right) - 1 = 1.01 - 1 = 0.01 \)
\( R_{\text{Quartal}} = 3 \cdot R = 3 \cdot 200 = 600 \, \text{€}. \)
\( R_n = R_{\text{Quartal}} \cdot q \cdot \frac{(q^{4n} - 1)}{q - 1} \)
\( R_n = 600 \cdot 1.01 \cdot \frac{(1.01^{80} - 1)}{1.01 - 1} \)
\( 1.01^{80} \approx 2.21964 \)
\( R_n = 600 \cdot 1.01 \cdot \frac{(2.21964 - 1)}{0.01} \)
\( R_n = 600 \cdot 1.01 \cdot \frac{1.21964}{0.01} \)
\( R_n = 600 \cdot 1.01 \cdot 121.964 \)
\( R_n \approx 600 \cdot 123.18364 \)
\( R_n \approx 73910.18 \, \text{€} \)
b)
\( R \cdot q - R_n \cdot (q - 1) = 5000 \cdot 1.01 - 73910.18 \cdot 0.01 \)
\( = 5050 - 739.1018 \)
\( = 4310.8982 \)
\( \frac{R \cdot q}{R \cdot q - R_n \cdot (q - 1)} = \frac{5050}{4310.8982} \approx 1.1714 \)
\( n = \frac{\ln(1.1714)}{\ln(1.01)} \)
\( n \approx \frac{0.1583}{0.00995} \approx 15.91 \)
\( \text{Anzahl der Jahre} = \frac{15.91}{4} \approx 3.98 \, \text{Jahre} \)
