REPEATING DECIMAL AS INFINITE GEOMETRIC SERIES

Problem 1 :

Express 0.37373737… as an infinite geometric series and find the fraction it represents.

Solution :

Let x = 0.37373737…

Writing the given repeating decimal as sum of smaller units of repeating decimals, we get

= 0.37 + 0.0037 + 0.000037 + …

= 37/100 + 37/10000 + 37/1000000 + …

= 37/100 + 37/100(1/100) + 37/100(1/100)² + …

a = 37/100 and r = 1/100

What fractions do these decimals represent?

Problem 2 :

0.52525252…

Solution :

Let x = 0.52525252…

= 0.52 + 0.0052 + 0.000052 + …

= 52/100 + 52/10000 + 52/1000000 + …

= 52/100 + 52/100(1/100) + 52/100(1/100)² + …

a = 52/100, r = 1/100

Problem 3 :

0.358358358…

Solution :

Let x = 0.358358358…

= 0.358 + 0.000358 + 0.000000358 + …

= 358/1000 + 358/1000000 + 358/100000000 + …

= 358/1000 + 358/1000(1/1000) + 358/1000(1/1000)² + …

a = 358/1000, r = 1/1000

  x = 358/999

So, the fraction is 358/999.

Problem 4 :

0.194949494…

Solution :

Let x = 0.194949494…

= 0.1 + 0.094 + 0.00094 + 0.0000094 + …

= 0.1 + 0.094[1 + 10^-2 + 10^-4 + …]

= 0.1 + 0.094[1/(1 – 10^-2)]

= 0.1 + 0.094[10²/99]

= 0.1 + 0.094[100/99]

= 1/10 + 94/1000 ×100/99

= 1/10 + 47/5 ×1/99

=1/10 + 47/495

= (1/10) × 99/99 + 47/495 × 2/2

= 99/990 + 94/990

= (99 + 94)/990

= 193/990

So, the fraction is 193/990.