Finding the Missing Number in a Number Series

Let's analyze the given number series to find the missing number: 14, 22, 37, 61, 96, ?

To find the pattern in this number series, we can look at the difference between consecutive terms.

Step 1: Calculate the First Level Differences

Find the difference between each term and the previous term:

• Difference between the 2nd and 1st term: $22 - 14 = 8$
• Difference between the 3rd and 2nd term: $37 - 22 = 15$
• Difference between the 4th and 3rd term: $61 - 37 = 24$
• Difference between the 5th and 4th term: $96 - 61 = 35$

The first level differences are: 8, 15, 24, 35.

Step 2: Calculate the Second Level Differences

Now, let's find the difference between the consecutive terms of the first level differences:

• Difference between the 2nd and 1st difference: $15 - 8 = 7$
• Difference between the 3rd and 2nd difference: $24 - 15 = 9$
• Difference between the 4th and 3rd difference: $35 - 24 = 11$

The second level differences are: 7, 9, 11.

Step 3: Identify the Pattern in the Second Level Differences

The sequence of second level differences (7, 9, 11) shows a clear pattern: these are consecutive odd numbers.

The next term in this sequence of second level differences should be the next odd number after 11, which is 13.

Step 4: Predict the Next First Level Difference

Based on the pattern, the next difference in the first level will be the last first level difference (35) plus the next second level difference (13):

Next first level difference $= 35 + 13 = 48$

Step 5: Calculate the Missing Number

The missing number in the series is found by adding the next first level difference (48) to the last term in the original series (96):

Missing number $= 96 + 48 = 144$

Summary of the Number Series Pattern

The pattern shows that the differences between consecutive terms increase by consecutive odd numbers (7, 9, 11, 13...). Therefore, the missing number is 144.

Revision Table: Number Series Analysis

Review the steps involved in solving number series questions by the difference method:

• Calculate differences between adjacent terms.
• If the first differences don't show a simple pattern, calculate differences between the differences (second level differences).
• Look for arithmetic progressions, geometric progressions, or other simple patterns in the differences.
• Once the pattern in differences is found, extrapolate to find the next difference.
• Use the next difference to find the missing term in the original series.

Additional Information: Types of Number Series

Number series problems often involve different types of patterns. Understanding these can help in solving questions quickly:

• Arithmetic Series: The difference between consecutive terms is constant. Example: 3, 6, 9, 12... (difference is 3)
• Geometric Series: Each term is found by multiplying the previous term by a constant ratio. Example: 2, 6, 18, 54... (ratio is 3)
• Difference Series: The differences between terms follow a pattern (like the one in this question).
• Mixed Series: Involve more than one pattern or operation (e.g., alternating addition and subtraction, or a combination of arithmetic and geometric progressions).
• Fibonacci Series: Each term is the sum of the two preceding terms (starting from 0 and 1 or 1 and 1). Example: 0, 1, 1, 2, 3, 5, 8...
• Square/Cube Series: Terms are squares or cubes of numbers, or related to them (e.g., $n^2$, $n^2+1$, $n^3$, $n^3-1$).

Practicing different types of series helps in quickly identifying the underlying pattern in competitive exams.