§8.3 Geometric sums

Introduction

Finite Geometric Series

Infinite Geometric Series

Let's explore sums of the form

. Observe that we can think of the first term, $1$, as $(\frac{1}{2})^0$, and since the powers of $\frac{1}{2}$ include every whole number from $0$ to $n-1$, there are exactly $n$ terms in the sum given by $S_n$.

Note that $S_1 = 1$, $S_2 = S_1 + \frac{1}{2} = 1 + \frac{1}{2} = \frac{3}{2}$, and $S_3 = S_2 + \frac{1}{4} = \frac{3}{2} + \frac{1}{4} = \frac{7}{4}$. Using the fact that each subsequent value of $S_n$ can be computed by adding one additional term to the preceding sum, complete Table with the exact (fractional) value of each sum.

$n=1$ | $S_1 = 1$

$n=2$ | $S_2 = \frac{3}{2}$

$n=3$ | $S_3 = \frac{7}{4}$

$n=4$ | $S_4 =$

$n=5$ | $S_5 =$

$n=6$ | $S_6 =$

$n=7$ | $S_7 =$

Multiply both sides of Equation by $r = \frac{2}{5}$. Write the new equation in the form

.

Now subtract Equation from Equation, and explain why it follows that

.

Solve Equation for $S_n$ to find a simple formula for $S_n$ that does not involve adding $n$ terms.

How would your work above change if instead of the original geometric sum $S_n$, we considered the situation with $a = 7$,

?

Use the shortcut formula in Equation to find the exact value of the finite geometric series

.

Use the shortcut formula in Equation to find the exact value of the finite geometric series

.

Use the shortcut formula in Equation to find the exact value of the finite geometric series

.

Can we find the value of

, where the sum never terminates?

$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \cdots$

$4 - 2 + 1 - \frac{1}{2} + \frac{1}{4} - \cdots$

$2 + \frac{8}{3} + \frac{32}{9} + \frac{128}{27} + \cdots$

$\sum_{k = 0}^{\infty} 5 \cdot \left( \frac{3}{4} \right)^k$

$\sum_{k = 1}^{\infty} -2 \cdot \left( -\frac{2}{3} \right)^k$