Finding the Missing Number in a Series

The question asks us to find the number that replaces the question mark (?) in the given series: 102, 110, 126, 150, 182, ?

To solve this type of number series question, we usually look for a pattern in the differences between consecutive terms, or a pattern related to multiplication, division, squares, cubes, or other mathematical operations.

Step-by-Step Analysis of the Series Pattern

Let's find the difference between each consecutive term:

• Difference between the 2nd and 1st term: $110 - 102$
• Difference between the 3rd and 2nd term: $126 - 110$
• Difference between the 4th and 3rd term: $150 - 126$
• Difference between the 5th and 4th term: $182 - 150$

Calculating these differences, we get:

• $110 - 102 = 8$
• $126 - 110 = 16$
• $150 - 126 = 24$
• $182 - 150 = 32$

Let's list these differences to see if there's a pattern:

Looking at the differences (8, 16, 24, 32), we can see a clear pattern. These numbers are multiples of 8 ($8 \times 1$, $8 \times 2$, $8 \times 3$, $8 \times 4$). This indicates that the differences are increasing by 8 each time. This sequence of differences is an arithmetic progression with a common difference of 8.

Following this pattern, the next difference in the series should be the next multiple of 8 after 32, which is $8 \times 5 = 40$. Alternatively, using the arithmetic progression of differences, the next difference is $32 + 8 = 40$.

To find the next number in the main series, we add this next difference (40) to the last term of the series (182).

Next term = Last term + Next difference

Next term = $182 + 40$

Next term = $222$

So, the number that replaces the question mark is 222.

Verifying the Pattern

The series can be represented as:

• 1st term: 102
• 2nd term: $102 + 8 = 110$
• 3rd term: $110 + 16 = 126$ (where $16 = 8 + 8$)
• 4th term: $126 + 24 = 150$ (where $24 = 16 + 8$)
• 5th term: $150 + 32 = 182$ (where $32 = 24 + 8$)
• 6th term: $182 + 40 = 222$ (where $40 = 32 + 8$)

The pattern holds true, confirming that the next number is 222.

Revision Table: Number Series Concepts

Additional Information: Solving Number Series Problems

When tackling number series problems, here are some common strategies and patterns to look out for:

• Check Differences: Calculate the difference between consecutive terms. If the differences form a recognizable pattern (constant, arithmetic series, geometric series, squares, cubes, etc.), you've likely found the rule.
• Check Double Differences: If the first differences don't show a clear pattern, calculate the differences between the differences (second differences). Sometimes, the second differences will have a constant value or a simple pattern.
• Check Ratios: Calculate the ratio between consecutive terms (term N / term N-1). This is useful for geometric series.
• Look for Squares, Cubes, etc.: The terms might be related to squares ($1, 4, 9, 16, ...$), cubes ($1, 8, 27, 64, ...$), or these values plus or minus a constant or a sequence of numbers.
• Alternating Patterns: Sometimes, the pattern alternates between two different rules or applies only to alternate terms.
• Combination of Operations: The pattern might involve a combination of operations, such as multiply by a number and then add or subtract another number ($a_n = a_{n-1} \times x + y$).
• Look for Prime Numbers, Fibonacci Sequence, etc.: Some series are based on known mathematical sequences like prime numbers (2, 3, 5, 7, 11, ...) or the Fibonacci sequence (1, 1, 2, 3, 5, 8, ... where each term is the sum of the two preceding ones).

Systematic calculation of differences is a good starting point for most series problems, as demonstrated in solving this question.