Understanding the Number Series Pattern

We are given the number series: 1, 5, 11, 19, 29, ____. To find the missing number, we need to identify the pattern or rule that governs the sequence.

Let's look at the differences between consecutive terms in the series:

• Difference between the 2nd and 1st term: $5 - 1 = 4$
• Difference between the 3rd and 2nd term: $11 - 5 = 6$
• Difference between the 4th and 3rd term: $19 - 11 = 8$
• Difference between the 5th and 4th term: $29 - 19 = 10$

The sequence of differences is 4, 6, 8, 10.

Identifying the Pattern in Differences

Now, let's look at the pattern in these differences:

• The difference between the second difference (6) and the first difference (4) is $6 - 4 = 2$.
• The difference between the third difference (8) and the second difference (6) is $8 - 6 = 2$.
• The difference between the fourth difference (10) and the third difference (8) is $10 - 8 = 2$.

We observe that the differences between consecutive terms of the original series form an arithmetic progression with a common difference of 2. This means the next difference in the sequence of differences (4, 6, 8, 10) should be 10 + 2 = 12.

Calculating the Missing Number

To find the missing number in the original series, we need to add the next difference (which is 12) to the last term of the series (which is 29).

Missing number = Last term + Next difference

Missing number = $29 + 12$

Missing number = $41$

Therefore, the next number in the series 1, 5, 11, 19, 29, ____ is 41.

The complete series is 1, 5, 11, 19, 29, 41.