exponential function

Question

P +Pr+(P +Pr)r = P(1+r)^2. Based on you work with exponential growth, interpret the three individual terms on the left side of this equation, and explain why their sum should equal the expression on the right side.

Answer 1

Hi Lorena T.,

Let's take your question in reverse order.

P + Pr + (P + Pr) r = (regrouping and extracting common factors) =

(P (1+r) )+ (P (1+r)) r = P (1+r) (1+r) Q.E.D.

Now the first part, explain the terms. It's the familiar old dissection of the incremental increase of a non-constant-slope function. If you haven't had this as calculus, you will soon.

P = present value at point x

Pr = incremental addition based on the growth rate **at x** (the starting point of the interval!)

(P + Pr) r = a compensation term for the increase in growth rate across the interval; it may be easier to see exactly why this works by re-writing the initial left side of the equation as:

P + Pr + (P + Pr) r = P + Pr + Pr + Pr^2 = P (1 + 2r + r^2) = P (1+r)^2

Do you see? The compensation term cleverly made up that 2nd Pr, so that you ended up with the binary expansion of (1 + r)^2 as an extractible factor.

-- Cheers, --Mr. d.

Answer 2

Hi Lorena!

P + Pr + (P+Pr)r

P(1+r) + P(1+r)r Factor out P

P(1+r) + P(1+r)r Notice that each term shares P(1+r) as a factor

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

P(1+r)² You have now arrived at the right side