Finding the Missing Term in a Number Series

Let's analyze the given number series to find the pattern and determine the missing term.

The series is: \(4, 6, 9, 13\frac{1}{2}, ______\)

First, let's write the mixed fraction as an improper fraction or a decimal to make calculations easier.

• \(13\frac{1}{2} = 13 + 0.5 = 13.5\)
• \(13\frac{1}{2} = \frac{13 \times 2 + 1}{2} = \frac{27}{2}\)

So the series can be written as: \(4, 6, 9, 13.5, ______\)

Identifying the Pattern in the Series

We can look at the difference between consecutive terms or the ratio between them.

Checking Differences:

• Difference between 6 and 4: \(6 - 4 = 2\)
• Difference between 9 and 6: \(9 - 6 = 3\)
• Difference between 13.5 and 9: \(13.5 - 9 = 4.5\)

The differences (2, 3, 4.5) do not form a simple arithmetic progression. This suggests the pattern might not be based on adding a constant or linearly increasing value.

Checking Ratios:

Let's check the ratio of each term to the previous term.

• Ratio of 6 to 4: \(\frac{6}{4} = 1.5\)
• Ratio of 9 to 6: \(\frac{9}{6} = 1.5\)
• Ratio of 13.5 to 9: \(\frac{13.5}{9} = \frac{135}{90} = \frac{27}{18} = \frac{3}{2} = 1.5\)

The ratio between consecutive terms is constant and equal to 1.5 (or \(\frac{3}{2}\)). This indicates that the series follows a geometric progression, where each term is obtained by multiplying the previous term by a constant factor (the common ratio).

Calculating the Missing Term

The pattern is to multiply the previous term by 1.5 (or \(\frac{3}{2}\)). The last given term is \(13.5\) or \(\frac{27}{2}\).

To find the missing term, we multiply the last term by 1.5:

Missing Term = \(13.5 \times 1.5\)

Calculation:

• \(13.5 \times 1.5 = (10 + 3 + 0.5) \times 1.5\)
• \(= 10 \times 1.5 + 3 \times 1.5 + 0.5 \times 1.5\)
• \(= 15 + 4.5 + 0.75\)
• \(= 19.5 + 0.75 = 20.25\)

Alternatively, using fractions:

Missing Term = \(\frac{27}{2} \times \frac{3}{2}\)

Missing Term = \(\frac{27 \times 3}{2 \times 2} = \frac{81}{4}\)

Converting the Result to Mixed Fraction

The calculated missing term is 20.25 or \(\frac{81}{4}\). Let's convert \(\frac{81}{4}\) into a mixed fraction to match the options.

\(\frac{81}{4} = 81 \div 4\)

• 81 divided by 4 is 20 with a remainder of 1.
• So, \(\frac{81}{4} = 20 + \frac{1}{4} = 20\frac{1}{4}\).

The missing term is \(20\frac{1}{4}\).

Comparing with Options

Let's compare our result \(20\frac{1}{4}\) with the given options:

• Option 1: \(22\frac{3}{4}\)
• Option 2: \(20\frac{1}{4}\)
• Option 3: \(19\)
• Option 4: \(17\frac{1}{2}\)

Our calculated missing term \(20\frac{1}{4}\) matches Option 2.

Conclusion: The Missing Term

Based on the pattern observed (multiplying by 1.5 or \(\frac{3}{2}\)), the missing term in the series \(4, 6, 9, 13\frac{1}{2}, ______\) is \(20\frac{1}{4}\).

Revision Table: Number Series Concepts

Additional Information: Solving Number Series Problems

Solving number series problems requires observing the relationship between the terms. Here are some common strategies:

• Check for Constant Difference: See if the same number is added or subtracted each time (Arithmetic Progression).
• Check for Constant Ratio: See if the same number is multiplied or divided each time (Geometric Progression).
• Check for Differences of Differences: If the first differences don't show a pattern, check the differences between those differences.
• Check for Product/Division Patterns: Sometimes terms are products or quotients of previous terms.
• Check for Squares/Cubes: Terms might be related to squares, cubes, or their roots.
• Check for Alternating Patterns: There might be two different patterns applied alternately to terms.
• Look for Prime Numbers, Fibonacci Sequence, etc.: Some series use famous mathematical sequences.
• Convert to Decimal/Fraction: As seen in this problem, converting fractions to decimals or vice-versa can sometimes reveal the pattern more clearly.

Practice is key to recognizing different types of number series patterns quickly during exams.