# Compound Interest

• Jun 27th 2010, 11:17 PM
jhonydeep3305
Compound Interest
How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
I get confused by the factored version P(1 + r)n. This does not help me to understand.

For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
• Jun 28th 2010, 12:02 AM
Prove It
Quote:

Originally Posted by jhonydeep3305
How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
I get confused by the factored version P(1 + r)n. This does not help me to understand.

For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.

I don't really understand what you mean...

But if it helps...

Compound interest can be written as a series of recursive simple interest functions with interest being charged on interest in each time period. This can then be converted to a closed form, which you know as the compound interest formula.

You should know the simple interest formula

$I = PrT$

so the amount accumulated is

$A = P + I$.

If we assume that we recalculate the simple interest every time period, then $T = 1$ and we can rewrite $A$ as a new principal for the next calculation.

So...

$P_1 = P_0 + I_0$

$= P_0 + P_0r$

$= P_0(1 + r)$.

$P_2 = P_1 + I_1$

$= P_1 + P_1r$

$= P_1(1 + r)$

$= P_0(1 + r)(1 + r)$

$= P_0(1 + r)^2$.

$P_3 = P_2 + I_2$

$= P_2 + P_2r$

$= P_2(1 + r)$

$= P_0(1 + r)^2(1 + r)$

$= P_0(1 + r)^3$.

$P_4 = P_3 + I_3$

$= P_3 + P_3r$

$= P_3(1 + r)$

$= P_0(1 + r)^3(1 + r)$

$= P_0(1 + r)^4$.

I think you can now see that after $n$ time periods, the amount accumulated equates to

$A = P(1 + r)^n$, with $I = A - P$.
• Jun 28th 2010, 04:33 AM
Wilmer
Quote:

Originally Posted by jhonydeep3305
How do I convert P(1 + r)3 and P( 1 + r)4, etc. to the expanded form?
I get confused by the factored version P(1 + r)n. This does not help me to understand.

For example, P(1 + r)2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.

To start, P(1 + r)^n means what \$P deposited today will be worth in n years at annual rate r%.

EXAMPLE: if \$1000 is deposited today, rate is 9%, what will it accumulate to over 3 years?
1000(1 + .09)^3 = 1295.03

The calculation is 1000 * 1.09 * 1.09 * 1.09 which results in 1295.03

> For example, P(1 + r)^2 equals (P + rP) + r(P + rP). Show how this occurs with three and four years.
I also don't understand what you're getting at here; seems irrelevant....
(1 + r)^2 = (1 + r) * (1 + r) = r^2 + 2r + 1; so P(r^2 + 2r + 1)