1.4.4 Sum to Infinity of Geometric Progressions

1.4.4 Sum to Infinity of Geometric Progressions

(G) Sum to Infinity of Geometric Progressions

S = a 1 r , 1 < r < 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) 2 3 , 2 9 , 2 27 , …..   
(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 = a 1 r = 2 1 1 2 = 4

(b)
2 3 , 2 9 , 2 27 , ..... a = 2 3 , r = 2 / 9 2 / 3 = 1 3 S = a 1 r S = 2 3 1 1 3 = 1

(c)
3 , 1 , 1 3 , ..... a = 3 , r = 1 3 S = a 1 r S = 3 1 1 3 = 3 2 / 3 = 9 2

(H) Recurring Decimal
Example of recurring decimal:
2 9 = 0.2222222222222..... 8 33 = 0.242424242424..... 41 333 = 0.123123123123.....

Recurring decimal can be changed to fraction using the sum to infinity formula:
S = a 1 r

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)
G P , a = 0.8 , r = 0.08 0.8 = 0.1 S = a 1 r S = 0.8 1 0.1 S = 0.8 0.9 S = 8 9 check using calculator 8 9 = 0.888888....

(b)

0.17171717 …..
= 0.17 + 0.0017 + 0.000017 + 0.00000017 + …..
G P , a = 0.17 , r = 0.0017 0.17 = 0.01 S = a 1 r S = 0.17 1 0.01 = 0.17 0.99 = 17 99 remember to check the answer using calculator

(c)
0.513513513…..
= 0.513 + 0.000513 + 0.000000513 + …..
G P , a = 0.513 , r = 0.00513 0.513 = 0.001 S = a 1 r S = 0.513 1 0.001 = 0.513 0.999 = 513 999 = 19 37

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