# Slow saving in banks

How long will it take to save € 9,000 by depositing € 200 at the beginning of each year at 2% interest?

Result

y =  32

#### Solution:

$q=2 \%=1 + \dfrac{ 2 }{ 100 }=1.02 \ \\ v=200 \ \text{eur} \ \\ r_{1}=v \cdot \ q=200 \cdot \ 1.02=204 \ \text{eur} \ \\ r_{2}=(r_{1}+v) \cdot \ q=(204+200) \cdot \ 1.02=\dfrac{ 10302 }{ 25 }=412.08 \ \\ r_{3}=(r_{2}+v) \cdot \ q=(412.08+200) \cdot \ 1.02=624.3216 \ \\ r_{4}=(r_{3}+v) \cdot \ q=(624.3216+200) \cdot \ 1.02 \doteq 840.808 \ \\ r_{5}=(r_{4}+v) \cdot \ q=(840.808+200) \cdot \ 1.02 \doteq 1061.6242 \ \\ r_{6}=(r_{5}+v) \cdot \ q=(1061.6242+200) \cdot \ 1.02 \doteq 1286.8567 \ \\ r_{7}=(r_{6}+v) \cdot \ q=(1286.8567+200) \cdot \ 1.02 \doteq 1516.5938 \ \\ r_{8}=(r_{7}+v) \cdot \ q=(1516.5938+200) \cdot \ 1.02 \doteq 1750.9257 \ \\ r_{9}=(r_{8}+v) \cdot \ q=(1750.9257+200) \cdot \ 1.02=1989.9442 \ \\ r_{10}=(r_{9}+v) \cdot \ q=(1989.9442+200) \cdot \ 1.02 \doteq 2233.7431 \ \\ r_{11}=(r_{10}+v) \cdot \ q=(2233.7431+200) \cdot \ 1.02 \doteq 2482.4179 \ \\ r_{12}=(r_{11}+v) \cdot \ q=(2482.4179+200) \cdot \ 1.02 \doteq 2736.0663 \ \\ r_{13}=(r_{12}+v) \cdot \ q=(2736.0663+200) \cdot \ 1.02 \doteq 2994.7876 \ \\ r_{14}=(r_{13}+v) \cdot \ q=(2994.7876+200) \cdot \ 1.02 \doteq 3258.6834 \ \\ r_{15}=(r_{14}+v) \cdot \ q=(3258.6834+200) \cdot \ 1.02 \doteq 3527.8571 \ \\ r_{16}=(r_{15}+v) \cdot \ q=(3527.8571+200) \cdot \ 1.02 \doteq 3802.4142 \ \\ r_{17}=(r_{16}+v) \cdot \ q=(3802.4142+200) \cdot \ 1.02 \doteq 4082.4625 \ \\ r_{18}=(r_{17}+v) \cdot \ q=(4082.4625+200) \cdot \ 1.02 \doteq 4368.1117 \ \\ r_{19}=(r_{18}+v) \cdot \ q=(4368.1117+200) \cdot \ 1.02 \doteq 4659.474 \ \\ r_{20}=(r_{19}+v) \cdot \ q=(4659.474+200) \cdot \ 1.02 \doteq 4956.6634 \ \\ r_{21}=(r_{20}+v) \cdot \ q=(4956.6634+200) \cdot \ 1.02 \doteq 5259.7967 \ \\ r_{22}=(r_{21}+v) \cdot \ q=(5259.7967+200) \cdot \ 1.02 \doteq 5568.9926 \ \\ r_{23}=(r_{22}+v) \cdot \ q=(5568.9926+200) \cdot \ 1.02 \doteq 5884.3725 \ \\ r_{24}=(r_{23}+v) \cdot \ q=(5884.3725+200) \cdot \ 1.02 \doteq 6206.0599 \ \\ r_{25}=(r_{24}+v) \cdot \ q=(6206.0599+200) \cdot \ 1.02 \doteq 6534.1811 \ \\ r_{26}=(r_{25}+v) \cdot \ q=(6534.1811+200) \cdot \ 1.02 \doteq 6868.8648 \ \\ r_{27}=(r_{26}+v) \cdot \ q=(6868.8648+200) \cdot \ 1.02 \doteq 7210.2421 \ \\ r_{28}=(r_{27}+v) \cdot \ q=(7210.2421+200) \cdot \ 1.02 \doteq 7558.4469 \ \\ r_{29}=(r_{28}+v) \cdot \ q=(7558.4469+200) \cdot \ 1.02 \doteq 7913.6158 \ \\ r_{30}=(r_{29}+v) \cdot \ q=(7913.6158+200) \cdot \ 1.02 \doteq 8275.8882 \ \\ r_{31}=(r_{30}+v) \cdot \ q=(8275.8882+200) \cdot \ 1.02 \doteq 8645.4059 \ \\ r_{32}=(r_{31}+v) \cdot \ q=(8645.4059+200) \cdot \ 1.02 \doteq 9022.314 \ \\ r_{33}=(r_{32}+v) \cdot \ q=(9022.314+200) \cdot \ 1.02 \doteq 9406.7603 \ \\ r_{34}=(r_{33}+v) \cdot \ q=(9406.7603+200) \cdot \ 1.02 \doteq 9798.8955 \ \\ r_{35}=(r_{34}+v) \cdot \ q=(9798.8955+200) \cdot \ 1.02 \doteq 10198.8734 \ \\ \ \\ \ \\ y=32$

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