Question

Select the number from the given options that can replace the question mark (?) in the following series.
4, 11, 26, ?, 120, 247

• A. 48
• B. 57
• C. 84
• D. 42

Hint:

When a series grows quickly but not exponentially, test patterns like "multiply by a constant and add/subtract a changing number" (e.g., \(ax+b\), where \(b\) changes arithmetically).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a sequence of numbers with one missing term. We need to find the pattern governing the series to determine the missing number.
Step 2: Key Formula or Approach:
There are several ways to find the pattern. We can check the difference between consecutive terms (first-level difference), the difference of the differences (second-level difference), or look for a pattern involving multiplication and addition/subtraction.
Step 3: Detailed Explanation:
Let's try the multiplication and addition approach, as the numbers are increasing quite rapidly.


To get from 4 to 11: \(4 \times 2 + 3 = 8 + 3 = 11\)
To get from 11 to 26: \(11 \times 2 + 4 = 22 + 4 = 26\)
A clear pattern emerges: To get the next term, multiply the current term by 2 and add a number that increases by 1 each time.
The pattern is \(T_n = T_{n-1} \times 2 + (n+1)\), where \(T_1 = 4\).
Let's apply this pattern to find the missing term (?):


The missing term is the 4th term in the series. It is obtained from the 3rd term (26).
Following the pattern, we should multiply by 2 and add 5.
Missing term = \(26 \times 2 + 5 = 52 + 5 = 57\)
So the missing term is 57.
Let's verify the pattern with the rest of the series:


From 57 to 120: \(57 \times 2 + 6 = 114 + 6 = 120\). This matches.
From 120 to 247: \(120 \times 2 + 7 = 240 + 7 = 247\). This also matches.
The pattern is consistent throughout the series.
Step 4: Final Answer:
The number that replaces the question mark is 57.