Finding the Missing Number in the Series

This question asks us to identify the pattern in the given number series and use it to find the missing number. The series is 9, 17, 33, 65, ?, 257. To solve this, we need to look at the relationship between consecutive terms in the number series.

Analyzing the Number Series Pattern

Let's examine the difference between consecutive terms:

• Difference between the second and first term: \(17 - 9 = 8\)
• Difference between the third and second term: \(33 - 17 = 16\)
• Difference between the fourth and third term: \(65 - 33 = 32\)

We can observe a pattern in the differences: 8, 16, 32. Each difference is double the previous difference (\(16 = 8 \times 2\), \(32 = 16 \times 2\)). This suggests the pattern involves doubling the difference at each step.

Another way to look at the pattern is the relationship between each term and the next. Let's try to find an operation that transforms one term into the next:

• From 9 to 17: \(9 \times 2 = 18\). \(18 - 1 = 17\). So, multiply by 2 and subtract 1.
• From 17 to 33: \(17 \times 2 = 34\). \(34 - 1 = 33\). This also follows the pattern.
• From 33 to 65: \(33 \times 2 = 66\). \(66 - 1 = 65\). This also follows the pattern.

The pattern appears to be: Each term is obtained by multiplying the previous term by 2 and then subtracting 1. Let's express this relationship:

If \(T_n\) is the n-th term, then \(T_{n+1} = T_n \times 2 - 1\).

Step-by-Step Calculation to Find the Missing Number

The given series is 9, 17, 33, 65, ?, 257.

Let the missing number be \(X\). \(X\) is the 5th term in the series, and 65 is the 4th term.

Using the identified pattern (\(T_{n+1} = T_n \times 2 - 1\)), we can find the missing number by applying the rule to the 4th term (65):

• Missing Number \(X\) = \(65 \times 2 - 1\)
• Missing Number \(X\) = \(130 - 1\)
• Missing Number \(X\) = \(129\)

Verification of the Pattern

Now, let's check if the pattern holds for the next term, using the calculated missing number (129) to get the last term (257):

• Next Term = \(129 \times 2 - 1\)
• Next Term = \(258 - 1\)
• Next Term = \(257\)

Since this matches the last number in the given series (257), our calculated missing number (129) is correct.

Summary of the Number Series Pattern

The pattern in this number series is to multiply the previous number by 2 and subtract 1 to get the next number. Let's visualize the steps:

Thus, the number that replaces the question mark (?) in the series is 129.

Revision Table: Number Series Concepts

Additional Information on Solving Number Series

Solving number series problems requires careful observation and pattern recognition. Here are some common strategies:

• Calculate differences between consecutive terms. See if there's a pattern in the differences (arithmetic, geometric, or another series).
• Calculate ratios between consecutive terms. Check if it's a geometric series.
• Look for patterns involving squares, cubes, prime numbers, or Fibonacci sequence.
• Consider mixed operations (addition, subtraction, multiplication, division combined). For example, \(T_{n+1} = T_n \times a \pm b\) or \(T_{n+1} = T_n \div a \pm b\).
• Check for alternating patterns (e.g., add a number, then subtract a number, then add again).
• Look for patterns related to the position of the term in the series (e.g., \(T_n = n^2 + 1\)).
• Sometimes, the pattern involves adding or subtracting terms from earlier in the series (e.g., Fibonacci: next term is sum of previous two).

Practice with different types of number series helps in quickly identifying the underlying logic.