In the series 5, 19, 49, 101, 181, 295, ?, find the missing number by analysing the pattern in the changing differences between consecutive terms.

Introduction / Context:
This question gives a series of numbers that grow rapidly: 5, 19, 49, 101, 181, 295, ?. The increments between terms are not constant, and they themselves appear to increase in a structured way. This type of problem usually involves examining both the first and second differences to reveal a smooth underlying pattern.

Given Data / Assumptions:
The known terms in the series are:

• 5
• 19
• 49
• 101
• 181
• 295
• ?

We assume that a consistent rule dictates the growth of the first differences, which can be captured by looking at the second differences between the terms.

Concept / Approach:
The general strategy is:

• First, compute the differences between consecutive terms (first differences).
• Then, compute the differences between these differences (second differences).
• If the second differences follow a simple pattern, we can extend that pattern to obtain the next first difference and therefore the next term.

Step-by-Step Solution:
Step 1: Compute first differences.19 − 5 = 14.49 − 19 = 30.101 − 49 = 52.181 − 101 = 80.295 − 181 = 114.So the first differences are 14, 30, 52, 80 and 114.Step 2: Compute second differences between these first differences.30 − 14 = 16.52 − 30 = 22.80 − 52 = 28.114 − 80 = 34.Step 3: Observe that the second differences are 16, 22, 28, 34, which increase by 6 each time.Step 4: Following this pattern, the next second difference should be 34 + 6 = 40.Step 5: The next first difference should therefore be 114 + 40 = 154.Step 6: Add this to the last known term: 295 + 154 = 449.

Verification / Alternative check:
We can rebuild the entire series using these rules. Starting from 5, add first differences 14, 30, 52, 80, 114 and 154, where each first difference grows by second differences 16, 22, 28, 34 and 40. This exactly reproduces the known terms and leads logically to 449 as the next term. The pattern is smooth and consistent, with second differences forming a simple arithmetic progression.

Why Other Options Are Wrong:
Values 401, 351 and 501 do not satisfy the established pattern of second differences increasing by 6. If we plugged any of them into the series, the computed next first differences and second differences would break the smooth 16, 22, 28, 34, 40 structure. Therefore these values are inconsistent with the revealed rule.

Common Pitfalls:
Many learners stop after calculating first differences and feel that they are irregular. For more advanced series, it is essential to examine second differences as well. Recognising that the second differences themselves form a simple arithmetic progression is the key insight for harder series questions like this one.

Final Answer:
The missing number that continues the pattern correctly is 449.