Number Series Logic Explained

The question asks us to identify the pattern in the given number series and find the missing term represented by the question mark (?). The series is: 4, 7, 13, 24, 42, ?

Analyzing the Series Pattern

To find the missing number, let's examine the differences between consecutive terms:

• Difference between the 2nd and 1st term: $7 - 4 = 3$
• Difference between the 3rd and 2nd term: $13 - 7 = 6$
• Difference between the 4th and 3rd term: $24 - 13 = 11$
• Difference between the 5th and 4th term: $42 - 24 = 18$

The sequence of differences is 3, 6, 11, 18.

Identifying the Pattern in Differences

Now, let's look at the differences between these differences (the second-order differences):

• Difference between the 2nd and 1st difference: $6 - 3 = 3$
• Difference between the 3rd and 2nd difference: $11 - 6 = 5$
• Difference between the 4th and 3rd difference: $18 - 11 = 7$

The second-order differences are 3, 5, 7. This sequence is an arithmetic progression, where each term increases by 2.

Calculating the Next Term

Following the pattern of the second-order differences:

• The next second-order difference should be $7 + 2 = 9$.
• This means the next difference in the original series (the first difference) should be the previous difference plus this new second-order difference: $18 + 9 = 27$.
• Finally, the missing term in the original series is obtained by adding this next difference to the last known term: $42 + 27 = 69$.

Therefore, the number that should come in place of the question mark (?) is 69.