Number Series Pattern Analysis: 31, 32, 36, ?, 61, 86

We need to find the missing number in the given series: 31, 32, 36, ?, 61, 86.

Let's examine the differences between consecutive terms to identify a pattern:

• Difference between the second and first term: $32 - 31 = 1$
• Difference between the third and second term: $36 - 32 = 4$

The differences observed are 1 and 4. We can recognize these numbers as perfect squares:

• $1 = 1^2$
• $4 = 2^2$

This suggests that the difference between consecutive terms might be increasing squares of consecutive integers.

Let's test this hypothesis:

• First term = 31
• Second term = First term + $1^2$ = $31 + 1 = 32$
• Third term = Second term + $2^2$ = $32 + 4 = 36$
• Missing term (Fourth term) = Third term + $3^2$ = $36 + 9 = 45$

Now, let's verify if this pattern holds for the remaining terms:

• Fifth term = Missing term + $4^2$ = $45 + 16 = 61$
• Sixth term = Fifth term + $5^2$ = $61 + 25 = 86$

The calculated terms match the given series (31, 32, 36, 45, 61, 86). Therefore, the pattern of adding consecutive squares ($1^2, 2^2, 3^2, 4^2, 5^2, ...$) is correct.

The missing number in the series is 45.