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# 5.4.4 Sum to Infinity of Geometric Progressions

5.4.4 Sum to Infinity of Geometric Progressions

(G) Sum to Infinity of Geometric Progressions

$\overline{)\text{}S\infty =\frac{a}{1-r},\text{}-1

a = first term
r = common ratio
S∞ = sum to infinity

Example:
Find the sum to infinity of each of the following geometric progressions.
(a) 8, 4, 2, …
(b) $\frac{2}{3},\text{}\frac{2}{9},\text{}\frac{2}{27},\text{}\dots ..$
(c) 3, 1, , ….

Solution:
(a)
8, 4, 2, ….
a = 2, r = 4/8 = ½
S∞ = 8 + 4 + 2 + 0.5 + 0.25 + 0.125 + 0.0625 + 0.03125 + …..
$S\infty =\frac{a}{1-r}=\frac{2}{1-\frac{1}{2}}=4$

(b)
$\begin{array}{l}\frac{2}{3},\text{}\frac{2}{9},\text{}\frac{2}{27},\text{}.....\\ a=\frac{2}{3},\text{}r=\frac{2/9}{2/3}=\frac{1}{3}\\ S\infty =\frac{a}{1-r}\\ S\infty =\frac{\frac{2}{3}}{1-\frac{1}{3}}=1\end{array}$

(c)
$\begin{array}{l}3,\text{}1,\text{}\frac{1}{3},\text{}.....\\ a=3,\text{}r=\frac{1}{3}\\ S\infty =\frac{a}{1-r}\\ S\infty =\frac{3}{1-\frac{1}{3}}=\frac{3}{2/3}=\frac{9}{2}\end{array}$

(H) Recurring Decimal
Example of recurring decimal:
$\begin{array}{l}\frac{2}{9}=0.2222222222222.....\\ \\ \frac{8}{33}=0.242424242424.....\\ \\ \frac{41}{333}=0.123123123123.....\end{array}$

Recurring decimal can be changed to fraction using the sum to infinity formula:
$\overline{)\text{}S\infty =\frac{a}{1-r}\text{}}$

Example (Change recurring decimal to fraction)
Express each of the following recurring decimals as a fraction in its lowest terms.
(a) 0.8888 ...
(b) 0.171717...
(c) 0.513513513 ….

Solution:
(a)
0.8888 = 0.8 + 0.08 + 0.008 +0.0008 + ….. (recurring decimal)
$\begin{array}{l}GP,\text{}a=0.8,\text{}r=\frac{0.08}{0.8}=0.1\\ {S}_{\infty }=\frac{a}{1-r}\\ {S}_{\infty }=\frac{0.8}{1-0.1}\\ {S}_{\infty }=\frac{0.8}{0.9}\\ {S}_{\infty }=\frac{8}{9}\to \overline{)\begin{array}{l}\text{check using calculator}\\ \text{}\frac{8}{9}=0.888888....\end{array}}\end{array}$

(b)

0.17171717 …..
= 0.17 + 0.0017 + 0.000017 + 0.00000017 + …..
$\begin{array}{l}GP,\text{}a=0.17,\text{}r=\frac{0.0017}{0.17}=0.01\\ {S}_{\infty }=\frac{a}{1-r}\\ {S}_{\infty }=\frac{0.17}{1-0.01}=\frac{0.17}{0.99}=\frac{17}{99}\to \overline{)\begin{array}{l}\text{remember to check the}\\ \text{answer using calculator}\end{array}}\end{array}$

(c)
0.513513513…..
= 0.513 + 0.000513 + 0.000000513 + …..
$\begin{array}{l}GP,\text{}a=0.513,\text{}r=\frac{0.00513}{0.513}=0.001\\ {S}_{\infty }=\frac{a}{1-r}\\ {S}_{\infty }=\frac{0.513}{1-0.001}=\frac{0.513}{0.999}=\frac{513}{999}=\frac{19}{37}\end{array}$

### 1 thought on “5.4.4 Sum to Infinity of Geometric Progressions”

1. can u get me to open the geo progressions