Understanding the Decimal Number Series Pattern

The question asks us to identify the next decimal number in the given series: 10, 9.9, 9.89, 9.889, 9.8889, .........

Let's carefully examine the terms in the series to find the underlying pattern.

The given series is:

• Term 1: 10
• Term 2: 9.9
• Term 3: 9.89
• Term 4: 9.889
• Term 5: 9.8889

We can observe how the terms change from one to the next.

• From Term 1 to Term 2: $10 - 9.9 = 0.1$ (Term 2 is $10 - 0.1$)
• From Term 2 to Term 3: $9.9 - 9.89 = 0.01$ (Term 3 is $9.9 - 0.01$)
• From Term 3 to Term 4: $9.89 - 9.889 = 0.001$ (Term 4 is $9.89 - 0.001$)
• From Term 4 to Term 5: $9.889 - 9.8889 = 0.0001$ (Term 5 is $9.889 - 0.0001$)

The difference between consecutive terms is decreasing by a factor of 10 each time. The subtrahends are powers of 0.1: $0.1^1, 0.1^2, 0.1^3, 0.1^4, \ldots$.

Alternatively, we can look at the structure of the decimal part of the numbers starting from the second term:

• Term 2: 9.9 (one digit '9')
• Term 3: 9.89 (one '8', followed by one '9')
• Term 4: 9.889 (two '8's, followed by one '9')
• Term 5: 9.8889 (three '8's, followed by one '9')

This second perspective reveals a clear pattern in the digits after the decimal point. Starting from the second term, the number consists of '9.', followed by an increasing number of '8' digits, and finally ending with a single '9' digit. The number of '8' digits increases by one for each subsequent term.

• Term 2 has zero '8's after the decimal before the '9'.
• Term 3 has one '8' after the decimal before the '9'.
• Term 4 has two '8's after the decimal before the '9'.
• Term 5 has three '8's after the decimal before the '9'.

Following this pattern, the next term (Term 6) should have four '8' digits after the decimal point, followed by a single '9'.

Therefore, the next number in the series is 9.88889.

Let's verify this using the subtraction pattern as well:

The difference between Term 5 and Term 6 should be $0.0001 \times 0.1 = 0.00001$.

Next term = Term 5 - 0.00001 = $9.8889 - 0.00001 = 9.88889$.

Both patterns confirm that the next number is 9.88889.

Step-by-Step Calculation

1. Identify the pattern in the series differences: $10 - 9.9 = 0.1$, $9.9 - 9.89 = 0.01$, $9.89 - 9.889 = 0.001$, $9.889 - 9.8889 = 0.0001$.

2. Recognize that the difference is divided by 10 for each subsequent term.

3. The next difference will be $0.0001 / 10 = 0.00001$.

4. Subtract this difference from the last term of the series to find the next term: $9.8889 - 0.00001 = 9.88889$.

Analyzing the Options

Let's compare our calculated next term with the given options:

The number 9.88889 correctly follows the identified pattern of having four '8' digits after the decimal, ending with a '9'.

Conclusion on the Decimal Series

Based on the analysis of the pattern, the next decimal number in the series 10, 9.9, 9.89, 9.889, 9.8889, ... is 9.88889.

Revision Table: Decimal Series Patterns

Additional Information: Understanding Number Series

Number series questions are common in aptitude tests and evaluate your ability to identify patterns in a sequence of numbers. These patterns can involve arithmetic progressions (constant difference), geometric progressions (constant ratio), differences between terms forming a pattern, or specific constructions of terms based on their position or previous terms.

• Arithmetic Series: Each term is obtained by adding a constant value (common difference) to the previous term.
• Geometric Series: Each term is obtained by multiplying the previous term by a constant value (common ratio).
• Difference Series: The differences between consecutive terms form their own series with a recognizable pattern.
• Mixed Series: Combinations of different patterns.
• Decimal Series: Series involving decimal numbers, where the pattern might affect the whole number part, the decimal part, or the differences between terms.

For this specific decimal series, we saw a pattern both in the decreasing difference between terms and in the construction of the decimal digits themselves. Recognizing either pattern leads to the correct next number in the series.