# 1.8 Sum to Infinity of Geometric Progressions

1.8 Sum to Infinity of Geometric Progressions

(G) Sum to Infinity of Geometric Progressions

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)
(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)

(c)

(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:

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)

(b)
0.17171717 …..
= 0.17 + 0.0017 + 0.000017 + 0.00000017 + …..

(c)
0.513513513…..
= 0.513 + 0.000513 + 0.000000513 + …..