   SEARCH HOME Math Central Quandaries & Queries  Question from Kenneth: Hello: I have a question regarding the following compound interest and future value calculation. Year 1 P + rP equals balance after the first year. Year 2 (P + rP) + r(P + rP) equals balance after the second year. Year 3 ? equals balance after the third year. This question is in two parts. 1. What would follow for year three? 2. I know that a pattern will develop. What will it be so that I can determine the extended pattern for following years ? I thank you for your reply. Hi Kenneth,

In the expression P + rP the letter P stands for principal, the amount you have at the beginning of the year and r is the interest rate. Thus the expression P + rP says that at the end of the year you have the amount you had at the beginning of the year, plus the interest rate times the amount you had at the beginning of the year. Hence at the beginning of year 2 you had P + rP and thus at the end of year 2 you have (P + rP) + r(P + rP).

At the beginning of year 3 you have (P + rP) + r(P + rP) so what do you have at the end of year 3?

There definitely is a pattern developing and a little algebra makes the pattern clear. First of all P + rP has a common factor of P so I can write

P + rP = (1 + r)P.

Hence at the end of year 2 you have

(1 + r)P + r[(1 + r)P]

which has a common factor of (1 + r)P so at the end of the second year you have

(1 + r)P + r[(1 + r)P] = (1 + r)(1 + r)P = (1 + r)2P.

What do you have at the end of the third year?

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.