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?

Correct answer:

y =  32

Step-by-step explanation:

$$\begin{gathered} q=2\mathrm{\%}=1+{\displaystyle \frac{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={\displaystyle \frac{10302}{25}}=412\frac{2}{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 \end{gathered}$$

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