The Sum $S_n$ of $n$ terms of a G.P. with first term $T_1 = a$ and common ratio $r$ is given by $$ S_n = a + ar + ar^2 + ar^3 + \cdots + ar^{n – 1}$$

multiplying both sides of this equation by $r$ yields

$$rS_n = ar + ar^2 + ar^3 + \cdots + ar^{n – 1} + ar^n$$

subtracting the first equation from the second gives us

$$rS_n\mathbin{-}S_n = ar^n\mathbin{-}a$$

factorizing we get

$$S_n(r\mathbin{-}1) = a(r^n\mathbin{-}1)$$

thus

$$S_n = a\frac{r^n\mathbin{-}1}{r\mathbin{-}1} = a\frac{1\mathbin{-}r^n}{1\mathbin{-}r}$$

The last two expressions for $S_n$ are equivalent: we can convert any one to the other by multiplying the numerator and denominator by minus one. If $0 < \left| r \right| < 1$ then as $n$ tends to infinity $r^n$ tends to zero and we have following sum for the resulting infinite G,P.

$$S_\infty = a + ar + ar^2 + ar^3 + \cdots = \lim_{n\to\infty} a\frac{1\mathbin{-} r^n}{1\mathbin{-} r} = \frac{a}{1\mathbin{-} r}$$ 

Setting $a = 1$ gives us the following identity that we will find useful later on:

$$ 1 + r + r^2 + r^3 + r^4 + \cdots = \frac{1}{1\mathbin{-}r}$$

putting $r = -r$ gives a similarly useful twin identity:

$$ 1\mathbin{-}r + r^2\mathbin{-}r^3 + r^4\mathbin{-}\cdots = \frac{1}{1+r}$$

Of course both of these formulas only hold for 0 < |r| < 1

Exercises:

1. Express the repeating decimal 0.694444… as a fraction.

2. If a ball is dropped from a height $h$ then the rebound height is $eh$ where $0 \leq e \leq 1$ is called the coefficient of restitution. Using the (unrealistic) model where the ball bounces forever eventually coming to rest as time approaches infinity, find a formula for the total distance the ball travels. Examine the special cases where $e$ is 0, 0.5, 1.

3. At the start of every month you invest a dollar at 6% annual interest compounded monthly. How much money do you have at the end of ten years?

4. The midpoints of the sides of an equilateral triangle of area one unit square are joined dividing it into four equal equilateral triangles. The central triangle is colored white and the other three triangles are each divided in four equilateral triangles with the central one in each colored white. The process is continued ad infinitum resulting in an object called Sierpinski’s triangle (or gasket or sieve). Calculate the total area of all the white triangles.

5. Impurities are being removed from water in a multistage process. Show that if at every stage we remove a constant fraction $0 < r < 1$ of the impurities left behind in the previous stage then we can make the water as pure as we like. If $r = 0.001$ how many stages do we need for the purity to exceed 99%?