Finding the Missing Number in the Series

The question asks us to find the number that should replace the question mark (?) in the given number series: 240, 120, ?, 180, 360, 900.

To solve this number series problem, we need to identify the pattern that connects the terms in the sequence. Let's examine the relationship between consecutive numbers.

Analyzing the Number Series Pattern

Let's look at the operations that transform one term into the next:

• From 240 to 120: $120 \div 240 = 0.5$. So, $240 \times 0.5 = 120$.
• From 180 to 360: $360 \div 180 = 2$. So, $180 \times 2 = 360$.
• From 360 to 900: $900 \div 360 = 2.5$. So, $360 \times 2.5 = 900$.

We see that the known operations are multiplication by 0.5, 2, and 2.5. Let's list these multipliers in order:

0.5, ?, ?, 2, 2.5

The multipliers used are 0.5, followed by two unknown multipliers, then 2, and finally 2.5.

Let's look at the sequence of known multipliers: 0.5, 2, 2.5. The difference between the last two is $2.5 - 2 = 0.5$. This suggests that the multipliers might be increasing in steps of 0.5.

Let's assume the multipliers follow a pattern starting from 0.5 and increasing by 0.5 for each subsequent step:

Multiplier 1: 0.5

Multiplier 2: $0.5 + 0.5 = 1.0$

Multiplier 3: $1.0 + 0.5 = 1.5$

Multiplier 4: $1.5 + 0.5 = 2.0$ (This matches the observed multiplier from 180 to 360)

Multiplier 5: $2.0 + 0.5 = 2.5$ (This matches the observed multiplier from 360 to 900)

So, the sequence of multipliers seems to be 0.5, 1.0, 1.5, 2.0, 2.5.

Applying the Pattern to Find the Missing Term

Now, let's apply this pattern of multipliers to the given number series:

• Term 1: 240
• Term 2: Term 1 $\times$ Multiplier 1 = $240 \times 0.5 = 120$ (This matches the second term in the series).
• Term 3 (the missing term ?): Term 2 $\times$ Multiplier 2 = $120 \times 1.0 = 120$.
• Term 4: Term 3 $\times$ Multiplier 3 = $120 \times 1.5$. Let's calculate: $120 \times 1.5 = 120 \times (1 + 0.5) = 120 \times 1 + 120 \times 0.5 = 120 + 60 = 180$. (This matches the fourth term in the series).
• Term 5: Term 4 $\times$ Multiplier 4 = $180 \times 2.0 = 360$ (Matches the fifth term).
• Term 6: Term 5 $\times$ Multiplier 5 = $360 \times 2.5$. Let's calculate: $360 \times 2.5 = 360 \times (2 + 0.5) = 360 \times 2 + 360 \times 0.5 = 720 + 180 = 900$. (Matches the sixth term).

The pattern holds true for the entire series when the missing term is 120.

Result

The number that replaces the question mark (?) is 120.

Let's summarize the series and the multipliers in a table:

The missing number is indeed 120.

Number Series Revision Table

Additional Information on Number Series Patterns

Number series questions are common in reasoning and aptitude tests. They evaluate your ability to identify logical patterns and rules. Here are a few types of patterns you might encounter:

• Arithmetic Series: Each term is obtained by adding or subtracting a constant value to the previous term (e.g., 2, 4, 6, 8...).
• Geometric Series: Each term is obtained by multiplying or dividing the previous term by a constant value (e.g., 3, 6, 12, 24...).
• Difference Series: The difference between consecutive terms follows a pattern itself, which might be an arithmetic series, a geometric series, or another type of series.
• Mixed Series: The pattern involves a combination of operations (e.g., multiply by 2, then add 1; multiply by 3, then subtract 2).
• Fibonacci Series: Each term is the sum of the two preceding terms (e.g., 0, 1, 1, 2, 3, 5, 8...).
• Square or Cube Series: The terms are squares or cubes of natural numbers, or relate to them (e.g., 1, 4, 9, 16... or 1, 8, 27, 64...).

In the series 240, 120, ?, 180, 360, 900, we saw a pattern where the multiplier changes consistently. This highlights that patterns can be complex and involve sequential operations with evolving parameters.