Finding the Missing Number in a Series: 6, 24, 54, 96, 150, ?

This question asks us to identify the pattern in the given number series and determine the next number that replaces the question mark. The series is 6, 24, 54, 96, 150, ?. Let's explore the relationships between consecutive numbers to find the underlying rule.

Analyzing the Differences Between Terms

One common approach to solving number series problems is to look at the differences between consecutive terms.

• Difference between the 2nd and 1st term: $24 - 6 = 18$
• Difference between the 3rd and 2nd term: $54 - 24 = 30$
• Difference between the 4th and 3rd term: $96 - 54 = 42$
• Difference between the 5th and 4th term: $150 - 96 = 54$

The first differences form a new series: 18, 30, 42, 54. Let's look at the differences within this new series (the second differences).

• Difference between the 2nd and 1st first difference: $30 - 18 = 12$
• Difference between the 3rd and 2nd first difference: $42 - 30 = 12$
• Difference between the 4th and 3rd first difference: $54 - 42 = 12$

We observe a constant second difference of 12. This indicates that the original series is likely based on a quadratic pattern.

Following this pattern, the next first difference should be $54 + 12 = 66$.

Therefore, the next term in the original series would be the last term plus this next first difference: $150 + 66 = 216$.

Identifying a Direct Pattern

Let's look at the numbers in the series again: 6, 24, 54, 96, 150.

We can try to express each term using its position in the series (n = 1, 2, 3, ...).

• 1st term (n=1): 6
• 2nd term (n=2): 24
• 3rd term (n=3): 54
• 4th term (n=4): 96
• 5th term (n=5): 150

Let's consider common mathematical operations involving the position number, such as squaring it ($n^2$).

• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$

Now, let's compare these squares to the terms in the series:

• Term 1: 6 vs $1^2 = 1$
• Term 2: 24 vs $2^2 = 4$
• Term 3: 54 vs $3^2 = 9$
• Term 4: 96 vs $4^2 = 16$
• Term 5: 150 vs $5^2 = 25$

Notice that each term in the series appears to be a multiple of the corresponding square. Let's divide each term by the square of its position:

• Term 1: $6 / 1^2 = 6 / 1 = 6$
• Term 2: $24 / 2^2 = 24 / 4 = 6$
• Term 3: $54 / 3^2 = 54 / 9 = 6$
• Term 4: $96 / 4^2 = 96 / 16 = 6$
• Term 5: $150 / 5^2 = 150 / 25 = 6$

This reveals a consistent pattern: each term in the series is equal to the square of its position number multiplied by 6. The formula for the nth term ($T_n$) is $T_n = n^2 \times 6$.

Calculating the Next Term

We need to find the 6th term in the series (n=6). Using the pattern $T_n = n^2 \times 6$:

$T_6 = 6^2 \times 6$

$T_6 = 36 \times 6$

$T_6 = 216$

So, the next number in the series is 216.

Comparing with Options

Let's compare our result with the given options:

The calculated number, 216, matches Option 1.

Revision Table: Number Series Pattern

Additional Information on Number Series Problems

Number series questions test your ability to find mathematical patterns. Common patterns include:

• Arithmetic Series: A constant difference between consecutive terms.
• Geometric Series: A constant ratio between consecutive terms (multiplication or division).
• Difference Series: Looking at the differences between terms (first differences, second differences, etc.). If the second difference is constant, it's a quadratic series.
• Based on Squares or Cubes: Terms related to $n^2$, $n^3$, $n^2 \pm c$, $n^3 \pm c$, $n^2 \times c$, etc.
• Alternating Series: The pattern alternates between two different rules.
• Fibonacci-like Series: Each term is the sum of the previous two terms (or a variation).

To solve number series problems effectively:

• Look at the differences between terms.
• Look at the ratios between terms.
• Consider squares, cubes, and other powers of the term number.
• Try combinations of operations (e.g., multiply and add).
• If the pattern isn't obvious, calculate differences repeatedly until a constant difference is found or a simpler pattern emerges.