Expert: Abe Mantell - 8/25/2010

Question
QUESTION: 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.

What would follow for year three?



ANSWER: The following year it will be previous amount plus r times
the previous amount.  So, the 3rd year will be
(P + rP) + r(P + rP) + r[(P + rP) + r(P + rP)]

However, notice that these can be simplified:
End of Year 1: P+rP =P(1+r)
End of Year 2: (P+rP)+r(P + rP)=P(1+r)+Pr(1+r)=(P+Pr)(1+r)=P(1+r)^2
End of Year 3 we'd get P(1+r)^3
...
End of Year n: P(1+r)^n

Abe


---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for the reply.

Would the fourth year be as follows:

(P + rP) + r(P + rP) + (P + rP) + r(P + rP) + r[(P + rP) + r(P + rP) + (P + rP) + r(P + rP)]

Fifth year:

(P + rP) + r(P + rP) + (P + rP) + r(P + rP) + (P + rP) + r(P + rP) + (P + rP) + r(P + rP) + r[(P + rP) + r(P + rP) + (P + rP) + r(P + rP) + (P + rP) + r(P + rP) + (P + rP) + r(P + rP)]

If so, I can understand why P(1+r)^n is used.

I thank you for your follow-up reply.




ANSWER: End of 4th year:
(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]+r[(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]]
=P(1+r)^4

End of 5th year:
(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]+r[(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]]
+r[(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]+r[(P+rP)+r(P+rP)+r[(P+rP)+r(P+rP)]]]=P(1+r)^5



---------- FOLLOW-UP ----------

QUESTION: Hello:

I want to thank you for your reply.  

Would it be incorrect to use brackets with the end of two year calculation as in (P + rP) + r[(P + rP)]?
It would appear to be consistent with the other calculations for more than two years.

And also, do you teach any on-line mathematics classes at the college where you teach?

I thank you for your follow-up reply.

Answer
Hello again Kenneth,

Brackets are OK, but not necessary where you placed them, the
parentheses already group the same terms!

As for online courses at my college...we offer some, but *I* do not
teach any.  I much prefer the face-to-face interaction.

Abe