Solving the Number Series: Find the Missing Term

This question asks us to find the number that should replace the question mark in the given series: 12, 25, 51, 103, 207, ?

To solve this, we need to identify the pattern or rule that connects the numbers in the sequence.

Analyzing the Number Series Pattern

Let's look at the relationship between consecutive terms in the series:

• From the first term (12) to the second term (25):
• \(25 = 12 \times 2 + 1\)
• From the second term (25) to the third term (51):
• \(51 = 25 \times 2 + 1\)
• From the third term (51) to the fourth term (103):
• \(103 = 51 \times 2 + 1\)
• From the fourth term (103) to the fifth term (207):
• \(207 = 103 \times 2 + 1\)

Identifying the Series Rule

The pattern is consistent across the known terms in the series. Each term is obtained by multiplying the previous term by 2 and then adding 1.

The rule can be written as: Term\(_{n}\) = Term\(_{n-1}\) \(\times\) 2 + 1

Calculating the Missing Number

Now we apply this rule to the last known term, which is 207, to find the next term in the series.

• The missing term is the term after 207.
• Using the rule: Missing Term = 207 \(\times\) 2 + 1
• Calculate the product: \(207 \times 2 = 414\)
• Add 1 to the product: \(414 + 1 = 415\)

So, the missing number in the series is 415.

Summary of the Pattern and Next Term

The completed series is 12, 25, 51, 103, 207, 415.

Revision Table: Understanding Number Series

Additional Information on Finding Number Patterns

Solving number series questions often involves trying out different mathematical operations between consecutive numbers. Here are some common approaches:

• Calculate the difference between consecutive terms. If the differences are constant, it's an arithmetic series. If the differences follow a pattern themselves (e.g., increasing by a constant), it's a difference series.
• Calculate the ratio between consecutive terms. If the ratio is constant, it's a geometric series.
• Look for patterns involving multiplication, division, addition, or subtraction, or a combination of these.
• Check for patterns involving squares, cubes, or other powers of numbers.
• Sometimes the pattern involves two alternating series within one sequence.
• In some cases, the next term is the sum or product of the previous two terms (like the Fibonacci sequence).

Practice with different types of series helps in quickly recognizing common patterns during exams.