Appendix A 209Table A-4. The mathematics of simple interestYear Amount At Interest Earned Amount At End
Beginning Of During Year Of Year
(t) Year (Ft)
0 - - P
1 P Pi P + Pi
= P (1 + i)
2 P (1 + i) Pi P (1+ i) + Pi
= P (1 + 2i)
3 P (1 + 2i) Pi P (1+ 2i) + Pi
= P (1 + 3i)
n P (1 + (n-1)i) Pi P (1+ (n-1)i) + Pi
= P (1 + ni)
Pi dollars ($1000.08 = $8) of interest. At the end of the year 1 the balance
in the account is obtained by adding P dollars (the original principal, $100)
plus Pi (the interest earned during year 1, $8) to obtain P+Pi ($100+$8 =
$108). Through algebraic manipulation, the end of year 1 balance can be
expressed mathematically as P(1+i) dollars ($1001.08 = $108).
The beginning of year 2 is the same point in time as the end of year
1 so the balance in the account is P(1+i) dollars ($108). During year 2 the
account again earns Pi dollars ($8) of interest, since under simple com-
pounding interest is paid only on the original principal amount P ($100).
Thus at the end of year 2, the balance in the account is obtained by adding
P dollars (the original principal) plus Pi (the interest from year 1) plus Pi
(the interest from year 2) to obtain P+Pi+Pi ($100+$8+$8 = $116). After
some algebraic manipulation, this can conveniently be written mathemat-
ically as P(1+2i) dollars ($100*1.16 = $116).
Table A-4 extends the above logic to year 3 and then generalizes
the approach for year n. If we return our attention to our original goal of
developing a formula for Fn that is expressed only in terms of the present
amount P, the annual interest rate i, and the number of years n, the above
development and Table A-4 results can be summarized as follows: