The first term of a geometric series is 5 - Leaving Cert Mathematics - Question c - 2010

To express the recurring decimal 0.1333... as an infinite geometric series, we can represent it as:

$0.1333... = 0.1 + 0.03333...$

The second term, 0.03333..., can be rewritten as:

$0.03333... = \frac{3}{100} + \frac{3}{1000} + \frac{3}{10000} + \ldots$

This series is a geometric series with:

• First term ( a = 0.1 )
• Common ratio ( r = 0.1 )

Thus, we can express the series as:

$0.1333... = 0.1 + \left( \frac{3}{100} \times \frac{1}{1 - 0.1} \right)$

Therefore, we simplify it to the form ( \frac{\text{numerator}}{\text{denominator}} ), where:

$0.1333... = \frac{4}{30} = \frac{2}{15}$

Here, ( a = 2 ) and ( b = 15 ), satisfying ( a, b \in \mathbb{N} ).