1.8 Sum to Infinity of Geometric Progressions

1.8 Sum to Infinity of Geometric Progressions

(G) Sum to Infinity of Geometric Progressions

  S= a 1r , 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 1r = 2 1 1 2 =4

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

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)
GP, a=0.8, r= 0.08 0.8 =0.1 S = a 1r S = 0.8 10.1 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 + …..
GP, a=0.17, r= 0.0017 0.17 =0.01 S = a 1r S = 0.17 10.01 = 0.17 0.99 = 17 99 remember to check the answer using calculator
(c)
0.513513513…..
= 0.513 + 0.000513 + 0.000000513 + …..
GP, a=0.513, r= 0.00513 0.513 =0.001 S = a 1r S = 0.513 10.001 = 0.513 0.999 = 513 999 = 19 37