Question

Select the number from among the given options that can replace the mark (?) in the following series:
7, 16, 41, 94, 249, ?

• A. 568
• B. 468
• C. 586
• D. 486

Hint:

In many standard exams, this series is presented with the 5th term as 251. In that case, the differences simply follow an alternating rule: $D_n = (D_{n-1} \times 3) - 2$ then $D_n = (D_{n-1} \times 2) + 3$, which also leads to 568.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This is a complex number series where each term is derived using alternating multiplication and a variable adjustment constant.

Step 2: Key Formula or Approach:
The series follows a pattern of $a_{n+1} = (a_n \times \text{multiplier}) \pm c_n$, where the multiplier alternates between 2 and 3.

Step 3: Detailed Explanation:
Let's breakdown the progression of the series:
• $7 \times 2 + 2 = 16$
• $16 \times 3 - 7 = 41$
• $41 \times 2 + 12 = 94$
• $94 \times 3 - 33 = 249$ The adjustments ($2, 7, 12, 33$) follow a sub-pattern:
• $7 = (2 \times 3) + 1$
• $12 = (7 \times 2) - 2$
• $33 = (12 \times 3) - 3$
• Next constant $c_5 = (33 \times 2) + 4 = 70$ Now, calculate the final term using the next multiplier (2) and the new constant (70): \[ ? = (249 \times 2) + 70 \] \[ ? = 498 + 70 = 568 \]

Step 4: Final Answer:
The number that replaces the question mark is 568.