Assignment 12 Spreadsheet in Mathematics Explorations
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 |
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.