Question

Find the missing number in the following numerical series:
4, 11, 25, 53, ?, 221

• A. 109
• B. 106
• C. 141
• D. 179

Hint:

Number series with rapid, near-doubling growth are often governed by a pattern of $x_n = 2 \cdot x_{n-1} \pm c$.
Checking the difference between the double of a term and the next term immediately reveals the constant value $c = 3$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This is a standard number series completion problem.
We need to find the mathematical rule that transforms each term of the series into the next, and use this rule to calculate the missing number.

Step 2: Key Formula or Approach:
Let the terms of the series be $x_1, x_2, x_3, \dots$
We calculate the differences and ratios between consecutive terms:
\[ x_2 - x_1 = 11 - 4 = 7 \] \[ x_3 - x_2 = 25 - 11 = 14 \] \[ x_4 - x_3 = 53 - 25 = 28 \] Notice that the differences are doubling: $7, 14, 28, \dots$
Alternatively, we can express the sequence recursively:
\[ x_n = 2 \cdot x_{n-1} + 3 \]

Step 3: Detailed Explanation:

• Verify the Recursive Pattern:
- Term 1 to Term 2:
\[ 4 \times 2 + 3 = 8 + 3 = 11 \]
- Term 2 to Term 3:
\[ 11 \times 2 + 3 = 22 + 3 = 25 \]
- Term 3 to Term 4:
\[ 25 \times 2 + 3 = 50 + 3 = 53 \]
This rule is consistent across all given terms.

• Calculate the Missing Term (Term 5):
Apply the rule to Term 4 ($53$):
\[ x_5 = 53 \times 2 + 3 = 106 + 3 = 109 \]

• Verify with the Final Term (Term 6):
Apply the rule to Term 5 ($109$):
\[ 109 \times 2 + 3 = 218 + 3 = 221 \] Since this matches the final term of the series, the missing number is confirmed as $109$.

Step 4: Final Answer:
The missing number in the series is 109.
Therefore, the correct option is (A).