Number Series Pattern Analysis

The question asks to identify the missing numbers in the given number series: $101, 110, 122, 137, ?, 176, ?$. We need to find the pattern governing the series.

Step 1: Calculate Differences Between Consecutive Terms

First, let's find the difference between each consecutive pair of numbers in the series:

• Difference between 110 and 101: $110 - 101 = 9$
• Difference between 122 and 110: $122 - 110 = 12$
• Difference between 137 and 122: $137 - 122 = 15$
• Difference between 176 and the next term (which we need to find first).

The differences observed are $9, 12, 15$.

Step 2: Identify the Pattern in the Differences

Observe the pattern in the differences calculated: $9, 12, 15$.

The difference between consecutive differences is constant:

• $12 - 9 = 3$
• $15 - 12 = 3$

This indicates that the differences are increasing by 3 each time. This is an arithmetic progression for the differences.

Step 3: Calculate the First Missing Number

Following the pattern, the next difference should be $15 + 3 = 18$. To find the first missing number (replacing the first '?'), add this difference to the last known term before it (137): $137 + 18 = 155$.

Step 4: Calculate the Second Missing Number

The number 176 is given after the first missing number. Let's verify the pattern:

The difference after 18 should be $18 + 3 = 21$. Add this difference to the number we found (155): $155 + 21 = 176$. This matches the series.

Now, find the difference after 21, which is $21 + 3 = 24$. To find the second missing number (replacing the second '?'), add this difference to 176: $176 + 24 = 200$.

Step 5: Final Answer

The numbers that replace the question marks are 155 and 200.

Therefore, the complete series is $101, 110, 122, 137, 155, 176, 200$.

The correct option is the one containing 155 and 200.