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:

x0=2000 x1=x0 1.05+200=2000 1.05+200=2300 x2=x1 1.05+200=2300 1.05+200=2615 x3=x2 1.05+200=2615 1.05+200=117834=2945.75 x4=x3 1.05+200=2945.75 1.05+200=3293.0375 x5=x4 1.05+200=3293.0375 1.05+2003657.6894 x6=x5 1.05+200=3657.6894 1.05+2004040.5738 x7=x6 1.05+200=4040.5738 1.05+2004442.6025 x8=x7 1.05+200=4442.6025 1.05+2004864.7327 x9=x8 1.05+200=4864.7327 1.05+2005307.9693 x10=x9 1.05+200=5307.9693 1.05+2005773.3678 x11=x10 1.05+200=5773.3678 1.05+2006262.0361 x12=x11 1.05+200=6262.0361 1.05+2006775.138 x13=x12 1.05+200=6775.138 1.05+2007313.8949 x14=x13 1.05+200=7313.8949 1.05+2007879.5896 x15=x14 1.05+200=7879.5896 1.05+2008473.5691 x16=x15 1.05+200=8473.5691 1.05+2009097.2475 x17=x16 1.05+200=9097.2475 1.05+2009752.1099 x18=x17 1.05+200=9752.1099 1.05+20010439.7154 x19=x18 1.05+200=10439.7154 1.05+20011161.7012 x20=x19 1.05+200=11161.7012 1.05+20011919.7862 x21=x20 1.05+200=11919.7862 1.05+20012715.7755 x22=x21 1.05+200=12715.7755 1.05+20013551.5643 x23=x22 1.05+200=13551.5643 1.05+20014429.1425 x24=x23 1.05+200=14429.1425 1.05+20015350.5997 x25=x24 1.05+200=15350.5997 1.05+20016318.1296 x26=x25 1.05+200=16318.1296 1.05+20017334.0361 x27=x26 1.05+200=17334.0361 1.05+20018400.7379 x28=x27 1.05+200=18400.7379 1.05+20019520.7748 x29=x28 1.05+200=19520.7748 1.05+20020696.8136 x30=x29 1.05+200=20696.8136 1.05+20021931.6543 x31=x30 1.05+200=21931.6543 1.05+20023228.237 x32=x31 1.05+200=23228.237 1.05+20024589.6488 x33=x32 1.05+200=24589.6488 1.05+20026019.1313 x34=x33 1.05+200=26019.1313 1.05+20027520.0878 x35=x34 1.05+200=27520.0878 1.05+20029096.0922 x36=x35 1.05+200=29096.0922 1.05+20030750.8968 x37=x36 1.05+200=30750.8968 1.05+20032488.4417 x38=x37 1.05+200=32488.4417 1.05+20034312.8637 x39=x38 1.05+200=34312.8637 1.05+20036228.5069 x40=x39 1.05+200=36228.5069 1.05+20038239.9323 x41=x40 1.05+200=38239.9323 1.05+20040351.9289 x42=x41 1.05+200=40351.9289 1.05+20042569.5253 n=42x_{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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