Assignment 12

Write up #5 Explore problems of growth, e.g. savings, interest compounded.

1. Suppose we invest \$1000.00 at an annual interest rate of 7% compouded yearly. Over a period of 25 years, what is the total interest?

 Year Total Interest 0 1000 0 1 1070 70 2 1144.9 74.9000000000001 3 1225.043 80.143 4 1310.79601 85.7530099999999 5 1402.5517307 91.7557207 6 1500.730351849 98.178621149 7 1605.78147647843 105.05112462943 8 1718.18617983192 112.40470335349 9 1838.45921242015 120.273032588234 10 1967.15135728957 128.692144869411 11 2104.85195229983 137.70059501027 12 2252.19158896082 147.339636660989 13 2409.84500018808 157.653411227257 14 2578.53415020125 168.689150013166 15 2759.03154071533 180.497390514087 16 2952.16374856541 193.132207850073 17 3158.81521096499 206.651462399579 18 3379.93227573253 221.117064767549 19 3616.52753503381 236.595259301278 20 3869.68446248618 253.156927452367 21 4140.56237486021 270.877912374033 22 4430.40174110043 289.839366240215 23 4740.52986297746 310.12812187703 24 5072.36695338588 331.837090408422 25 5427.43264012289 355.065686737012

Total interest = 5427.43- 1000.00 = \$ 4427.43

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From the results above, and after some investigation, we can come up with a formula of the total to be 1000(1 + 0.07)^ T where T is the number of years. Now we can compare the total interests when the compounding periods are more than one per year.

2. The following table shows total interests for quarterly , monthly and continuously. The general formula for compound interest is A(1+r/n)^(nt) where A is the initial amount(1000 in this case), r is the interest rate(7% here), n is the compounding period,and t is the time in years. For continuous compound interests, we use the formula Ae^(rt).

 Year Yearly Quarterly Monthly Continuously 0 1000 1000 1000 1000 1 1070 1071.85903128906 1072.29008085624 1072.50818125422 2 1144.9 1148.88178295593 1149.80601750267 1150.27379885723 3 1225.043 1231.43931494479 1232.92558747693 1233.67805995674 4 1310.79601 1319.92935120799 1322.05387788536 1323.12981233744 5 1402.5517307 1414.7781957558 1417.62525961399 1419.06754859326 6 1500.730351849 1516.4427863917 1520.10550425533 1521.96155561863 7 1605.78147647843 1625.41289602709 1629.99405406795 1632.31621995538 8 1718.18617983192 1742.21349218035 1747.82645603171 1750.6725002961 9 1838.45921242015 1867.40726602717 1874.17697186091 1877.61057926434 10 1967.15135728957 2001.59734318604 2009.66137669563 2013.75270747048 11 2104.85195229984 2145.43018929815 2154.93996011061 2159.76625378492 12 2252.19158896082 2299.59872439942 2310.72074406734 2316.36697678109 13 2409.84500018808 2464.84566108833 2477.76293349215 2484.32253338482 14 2578.53415020125 2641.96708257119 2656.88061629689 2664.45624192942 15 2759.03154071533 2831.81627782234 2848.94673087435 2857.65111806316 16 2952.16374856541 3035.30785233526 3054.89732040437 3064.854203293 17 3158.81521096499 3253.42213426815 3275.7360947039 3287.08120738312 18 3379.93227573254 3487.20989721106 3512.53932185373 3525.42148736538 19 3616.52753503381 3737.79742232628 3766.46107344125 3781.04338756878 20 3869.68446248618 4006.3919242494 4038.73884898218 4055.19996684468 21 4140.56237486021 4294.28736689029 4330.69960693232 4349.23514106274 22 4430.40174110043 4602.87069715188 4643.76623168153 4664.59027098813 23 4740.52986297746 4933.62852659803 4979.46446804725 5002.81122783359 24 5072.36695338588 5288.15429325945 5339.43035706314 5365.55597112198 25 5427.43264012289 5668.15593808017 5725.41820930147 5754.60267600573 Total 4427.43264012289 4668.15593808017 4725.41820930147 4754.60267600573

From the bottom row of this table, one can see that the more the compounding period, the bigger the total interest that our principal earns. Of course, when teaching this to the students for the first time, one has to spend a good portion of the time deriving the formulas that we used in the spreadsheets. The formulas are all exponential functions and great examples for teaching the applications of exponential functions.

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