What number should come next in the series 7, 4.5, 5.5, 12, 49, ?

Introduction / Context:
This is a number series question where you must identify the pattern connecting consecutive terms and then extend it to find the next term. The series 7, 4.5, 5.5, 12, 49, ? may not follow a simple arithmetic or geometric progression, so we look for a pattern in how each term is generated from the previous one.

Given Data / Assumptions:
Given series: 7, 4.5, 5.5, 12, 49, ?We must determine the next term in this sequence.All terms are real numbers and may involve multiplication and addition patterns.

Concept / Approach:
We examine how each term arises from the previous one. One useful strategy is to look for multiplicative relationships combined with small additions. After testing a few possibilities, we notice that each term is approximately the previous term multiplied by a factor that doubles each time, plus 1. The factor sequence itself forms a pattern: 0.5, 1, 2, 4, then 8 for the next step.

Step-by-Step Solution:
Step 1: From 7 to 4.5: 7 × 0.5 + 1 = 3.5 + 1 = 4.5.Step 2: From 4.5 to 5.5: 4.5 × 1 + 1 = 4.5 + 1 = 5.5.Step 3: From 5.5 to 12: 5.5 × 2 + 1 = 11 + 1 = 12.Step 4: From 12 to 49: 12 × 4 + 1 = 48 + 1 = 49.Step 5: Observe that the multipliers are 0.5, 1, 2 and 4, which double each time.Step 6: The next multiplier should therefore be 8, continuing this doubling pattern.Step 7: Apply this rule to 49: 49 × 8 + 1 = 392 + 1 = 393.Step 8: So the next term in the series is 393.

Verification / Alternative check:
Verify the pattern consistency: each step is previous term times a factor plus 1, and the factors follow the sequence 1/2, 1, 2, 4, 8. This internal pattern is smooth and consistent, which strongly indicates the intended rule. No other reasonable simple pattern fits all the given terms as neatly.

Why Other Options Are Wrong:
The values 396, 354 and 367 do not arise from applying the identified rule of multiplying by 8 and adding 1. For example, 49 × 8 + 1 equals 393 exactly, not any of the other numbers. Alternative ad hoc patterns would require changing the rule mid-series, which is not typical in standard test questions.

Common Pitfalls:
Students may initially try only arithmetic differences or ratios and not notice the combined multiply then add pattern. Some might also miss the factor sequence doubling from 0.5 to 1 to 2 to 4. It helps to systematically test both additive and multiplicative relationships rather than giving up after the simplest attempts.

Final Answer:
The next number in the series is 393.