Number Series Progression

The problem asks us to find the missing numbers in the sequence: 83, 86, 91, 98, ?, 122, ?. We need to determine the logic behind the sequence to fill in the question marks.

Pattern Identification Logic

We begin by finding the differences between the consecutive known numbers in the series:

• Difference between 86 and 83: $86 - 83 = 3$
• Difference between 91 and 86: $91 - 86 = 5$
• Difference between 98 and 91: $98 - 91 = 7$

The differences obtained are 3, 5, and 7. These are consecutive prime numbers, starting after the prime number 2.

The list of prime numbers begins: 2, 3, 5, 7, 11, 13, 17, ...

Based on this observation, the pattern involves adding consecutive prime numbers (starting from 3) to get the next term in the series.

Calculation of Missing Numbers

Let's apply this pattern to find the missing numbers:

1. Start with the term 98.
2. The next prime number in the sequence is 11. Add this to 98: $98 + 11 = 109$. This is the first missing number.
3. The next term given in the series is 122. Let's check the difference using the next prime number, 13: $109 + 13 = 122$. This matches the given term, confirming the pattern.
4. The next prime number in the sequence is 17. Add this to 122 to find the second missing number: $122 + 17 = 139$.

So the complete series is 83, 86, 91, 98, 109, 122, 139.

Series Completion Check

The calculated numbers 109 and 139 fit the established pattern of adding consecutive primes (3, 5, 7, 11, 13, 17). The series progresses logically based on this rule.