# Saving 9

An amount if $2000 is invested at an interest of 5% per month. if$ 200 is added at the beginning of each successive month but no withdrawals. Give an expression for the value accumulated after n months. After how many months will the amount have accumulated first exceed \$ 42000.

Result

n =  42

#### Solution:

$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=\dfrac{ 11783 }{ 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$

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