Finding the Missing Number in a Series

Let's analyze the given number series: 16, 24, 36, ?, 81. We need to find the number that replaces the question mark.

To solve a number series problem, we look for a mathematical pattern between consecutive terms. Common patterns include addition, subtraction, multiplication, division, or a combination of these, sometimes involving squares, cubes, or other sequences.

Let's examine the relationship between the terms:

• From 16 to 24: The difference is $24 - 16 = 8$. The ratio is $\frac{24}{16} = \frac{3}{2} = 1.5$.
• From 24 to 36: The difference is $36 - 24 = 12$. The ratio is $\frac{36}{24} = \frac{3}{2} = 1.5$.

We can see a consistent ratio of 1.5 between consecutive terms. This suggests the pattern is multiplication by 1.5 (or $\frac{3}{2}$). Let's assume this pattern continues throughout the series.

Applying the Multiplicative Pattern to Find the Missing Term

Following the pattern, the next term after 36 should be obtained by multiplying 36 by 1.5.

Calculation:

• Missing term = $36 \times 1.5$
• Missing term = $36 \times \frac{3}{2}$
• Missing term = $\frac{108}{2}$
• Missing term = $54$

So, the number replacing the question mark is 54.

Verifying the Pattern with the Last Term

Let's check if applying the same pattern to 54 gives us the last term, 81.

• Next term = $54 \times 1.5$
• Next term = $54 \times \frac{3}{2}$
• Next term = $\frac{162}{2}$
• Next term = $81$

This matches the last term in the series. Therefore, the identified pattern is correct.

The completed series is 16, 24, 36, 54, 81.

The number that replaces the question mark (?) in the given series is 54.

Revision Table: Number Series Concepts

Understanding different types of number series patterns is key to solving these problems.

Additional Information on Solving Number Series Problems

Here are some tips for tackling number series questions in competitive exams or tests:

• Look for simple arithmetic progressions (constant difference).
• Check for geometric progressions (constant ratio).
• Analyze the differences between consecutive terms – they might form a simpler series.
• Look for differences between the differences (second-order differences).
• Consider squares, cubes, square roots, or cube roots of the term numbers or the terms themselves.
• Check for alternating patterns.
• Sometimes the pattern involves operations on the digits of the numbers.
• Practice with various types of number series to recognize patterns quickly.
• Don't spend too much time on one question; if a pattern isn't obvious, move on and come back if time permits.

Solving number series requires keen observation and practice to identify the underlying mathematical rule.