Using the exponential growth and decay formula for compound interestI have what seems to be a rather simple question but one that is confusing me a lot.When looking at standard exponential growth/decay models(such as the decay of Carbon-14 etc),we can use the formula A = P ( 1 + r ) t in order to find things such as the half-life/rate.However, in these models, aren't the substances (such as Carbon-14) assumed to decay continuously?If so, why can we NOT use the formula P e r t ,when is this assumed to be the formula for continuous growth/decay?For instance, in a compound interest problem where interest is compoundedcontinuously, we would have to use the P e r t formula right?(We couldn't use the formula A ( 1 + r ) t

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Using the exponential growth and decay formula for compound interestI have what seems to be a rather simple question but one that is confusing me a lot.When looking at standard exponential growth/decay models(such as the decay of Carbon-14 etc),we can use the formula   A  =  P  (  1  +  r      )    t   in order to find things such as the half-life/rate.However, in these models, aren&#039;t the substances (such as Carbon-14) assumed to decay continuously?If so, why can we NOT use the formula   P      e          r      t      ,when is this assumed to be the formula for continuous growth/decay?For instance, in a compound interest problem where interest is compoundedcontinuously, we would have to use the   P      e          r      t       formula right?(We couldn&#039;t use the formula   A  (  1  +  r      )    t

Gracelyn Paul · Accepted Answer

Suppose that I am saving up for 20 years, with some sort of savings bond, and that the interest rate is locked in at 5% per year, for the whole 20 years. I am depositing 1 million dollars, and the compounding period is going to be one of the choices below.For each of the following compounding periods, and P = 1, 000, 000, what you get is the following:First column = Period of CompoundingSecond column = Periods per Year mThird column = Number of Periods n=mtFourth column = Interest per Period i=r/mFifth column = Amount   A  =  P  (  1  +  i      )    n        [                                        A            n            n            u            a            l            l            y                                    1                          20                          (          0.05                      /                    1          =          0.05          )                          2          ,          653          ,          297.71                                                  S            e            m            i            a            n            n            u            a            l            l            y                                    2                          40                          (          0.05                      /                    2          =          0.025          )                          2          ,          685          ,          063.84                                                  Q            u            a            r            t            e            r            l            y                                    4                          80                          (          0.05                      /                    4          =          0.0125          )                          2          ,          701          ,          484.94                                                  B            i            m            o            n            t            h            l            y                                    6                          120                          (          0.05                      /                    6          =          0.0833..          )                          2          ,          707          ,          041.49                                                  M            o            n            t            h            l            y                                    12                          240                          (          0.05                      /                    12          =          0.0416..          )                          2          ,          712          ,          640.29                                                  B            i            w            e            e            k            l            y                                    26                          520                          (          0.05                      /                    26          =          0.00192307...          )                          2          ,          715          ,          672.70                                                  W            e            e            k            l            y                                    52                          1040                          (          0.05                      /                    52          =          0.000961538..          )                          2          ,          716          ,          976.11                                                  D            a            i            l            y                                    360                          7200                          (          0.05                      /                    360          =          0.00013888..          )                          2          ,          718          ,          093.08                                                  H            o            u            r            l            y                                    (          360          ∗          24          )                          172800                          (          5.78703..          ∗                      10                          −              6              )                                                2          ,          718          ,          273.96                                                  M            i            n            u            t            e            l            y                                    (          360          ∗          24          ∗          60          )                          10368000                          (          9.64506..          ∗                      10                          −              6              )                                                2          ,          718          ,          281.70                                                  S            e            c            o            n            d            l            y                                    (          360          ∗          24          ∗          60          ∗          60          )                          622080000                          (          1.60751..          ∗                      10                          −              6              )                                                2          ,          718          ,          281.92                      ]  Now surely, the last three compounding periods are just fictional. No one, except possibly a mafia loan-shark, would compound interest hourly. They are printed here to prove a point: observe that as you go down the table, n is getting very large—but the amount, A, is going toward a fixed number. This fixed number is the value of the continuously compounded interest where m =   ∞.Before you continue, you should verify the arithmetic.Let’s verify the daily one together. As we said before, bankers believe that there are 360 days per year.We know that i = r/m and since r = 0.05 in this case, our calculator tells us that i = 0.05/360 = 0.0001388... .The principal is given to us as $ 1,000,000.00.All we need now is n, and n = m * t = (360)(20) = 7200. Finally, we have  A  =  P  (  1  +  i      )    n    A  =  1  ,  000  ,  000  (  1  +  0.00013888  ·  ·  ·      )          7200      A=(1,000,000)(2.71809⋅⋅⋅)A=2,718,093.08

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