Finding the Number Series Pattern

The question asks to identify the next number in the given sequence: 52, 56, 65, 81, 106, ?

To solve this, let's analyze the differences between consecutive terms:

• $56 - 52 = 4$
• $65 - 56 = 9$
• $81 - 65 = 16$
• $106 - 81 = 25$

Identifying the Pattern Logic

Observe the differences calculated: 4, 9, 16, 25. These numbers are perfect squares:

• $4 = 2^2$
• $9 = 3^2$
• $16 = 4^2$
• $25 = 5^2$

The pattern shows that the difference between terms increases by consecutive squares, starting from $2^2$.

Calculating the Next Term

Following the pattern, the next difference should be the next square, which is $6^2$.

• Next difference $= 6^2 = 36$

To find the missing number in the series, add this difference to the last known term (106):

• Missing number $= 106 + 36$
• Missing number $= 142$

Therefore, the correct option is 142.