Nasledujúceho 5164

Suma 2 000 USD sa investuje s úrokom 5 % mesačne. Ak sa na začiatku každého nasledujúceho mesiaca pridá 200 USD, ale žiadne výbery. Zadajte výraz pre hodnotu nahromadenú po n mesiacoch. Po koľkých mesiacoch nahromadená suma presiahne 42 000 $?

Správna odpoveď:

n =  42

Postup správneho riešenia:

$$\begin{gathered} x_0=2000 \\ {x}_1=x_0\cdot 1,05+200=2000\cdot 1,05+200=2300 \\ {x}_2=x_1\cdot 1,05+200=2300\cdot 1,05+200=2615 \\ {x}_3=x_2\cdot 1,05+200=2615\cdot 1,05+200=\frac{11783}{4}=2945\frac{3}{4}=2945,75 \\ {x}_4=x_3\cdot 1,05+200=2945,75\cdot 1,05+200=3293,0375 \\ {x}_5=x_4\cdot 1,05+200=3293,0375\cdot 1,05+200\doteq 3657,6894 \\ {x}_6=x_5\cdot 1,05+200=3657,6894\cdot 1,05+200\doteq 4040,5738 \\ {x}_7=x_6\cdot 1,05+200=4040,5738\cdot 1,05+200\doteq 4442,6025 \\ {x}_8=x_7\cdot 1,05+200=4442,6025\cdot 1,05+200\doteq 4864,7327 \\ {x}_9=x_8\cdot 1,05+200=4864,7327\cdot 1,05+200\doteq 5307,9693 \\ {x}_{10}=x_9\cdot 1,05+200=5307,9693\cdot 1,05+200\doteq 5773,3678 \\ {x}_{11}=x_{10}\cdot 1,05+200=5773,3678\cdot 1,05+200\doteq 6262,0361 \\ {x}_{12}=x_{11}\cdot 1,05+200=6262,0361\cdot 1,05+200\doteq 6775,138 \\ {x}_{13}=x_{12}\cdot 1,05+200=6775,138\cdot 1,05+200\doteq 7313,8949 \\ {x}_{14}=x_{13}\cdot 1,05+200=7313,8949\cdot 1,05+200\doteq 7879,5896 \\ {x}_{15}=x_{14}\cdot 1,05+200=7879,5896\cdot 1,05+200\doteq 8473,5691 \\ {x}_{16}=x_{15}\cdot 1,05+200=8473,5691\cdot 1,05+200\doteq 9097,2475 \\ {x}_{17}=x_{16}\cdot 1,05+200=9097,2475\cdot 1,05+200\doteq 9752,1099 \\ {x}_{18}=x_{17}\cdot 1,05+200=9752,1099\cdot 1,05+200\doteq 10439,7154 \\ {x}_{19}=x_{18}\cdot 1,05+200=10439,7154\cdot 1,05+200\doteq 11161,7012 \\ {x}_{20}=x_{19}\cdot 1,05+200=11161,7012\cdot 1,05+200\doteq 11919,7862 \\ {x}_{21}=x_{20}\cdot 1,05+200=11919,7862\cdot 1,05+200\doteq 12715,7755 \\ {x}_{22}=x_{21}\cdot 1,05+200=12715,7755\cdot 1,05+200\doteq 13551,5643 \\ {x}_{23}=x_{22}\cdot 1,05+200=13551,5643\cdot 1,05+200\doteq 14429,1425 \\ {x}_{24}=x_{23}\cdot 1,05+200=14429,1425\cdot 1,05+200\doteq 15350,5997 \\ {x}_{25}=x_{24}\cdot 1,05+200=15350,5997\cdot 1,05+200\doteq 16318,1296 \\ {x}_{26}=x_{25}\cdot 1,05+200=16318,1296\cdot 1,05+200\doteq 17334,0361 \\ {x}_{27}=x_{26}\cdot 1,05+200=17334,0361\cdot 1,05+200\doteq 18400,7379 \\ {x}_{28}=x_{27}\cdot 1,05+200=18400,7379\cdot 1,05+200\doteq 19520,7748 \\ {x}_{29}=x_{28}\cdot 1,05+200=19520,7748\cdot 1,05+200\doteq 20696,8136 \\ {x}_{30}=x_{29}\cdot 1,05+200=20696,8136\cdot 1,05+200\doteq 21931,6543 \\ {x}_{31}=x_{30}\cdot 1,05+200=21931,6543\cdot 1,05+200\doteq 23228,237 \\ {x}_{32}=x_{31}\cdot 1,05+200=23228,237\cdot 1,05+200\doteq 24589,6488 \\ {x}_{33}=x_{32}\cdot 1,05+200=24589,6488\cdot 1,05+200\doteq 26019,1313 \\ {x}_{34}=x_{33}\cdot 1,05+200=26019,1313\cdot 1,05+200\doteq 27520,0878 \\ {x}_{35}=x_{34}\cdot 1,05+200=27520,0878\cdot 1,05+200\doteq 29096,0922 \\ {x}_{36}=x_{35}\cdot 1,05+200=29096,0922\cdot 1,05+200\doteq 30750,8968 \\ {x}_{37}=x_{36}\cdot 1,05+200=30750,8968\cdot 1,05+200\doteq 32488,4417 \\ {x}_{38}=x_{37}\cdot 1,05+200=32488,4417\cdot 1,05+200\doteq 34312,8637 \\ {x}_{39}=x_{38}\cdot 1,05+200=34312,8637\cdot 1,05+200\doteq 36228,5069 \\ {x}_{40}=x_{39}\cdot 1,05+200=36228,5069\cdot 1,05+200\doteq 38239,9323 \\ {x}_{41}=x_{40}\cdot 1,05+200=38239,9323\cdot 1,05+200\doteq 40351,9289 \\ {x}_{42}=x_{41}\cdot 1,05+200=40351,9289\cdot 1,05+200\doteq 42569,5253 \\ n=42 \end{gathered}$$

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