The question asks us to find the missing number in the sequence: 57, 69, 88, 112, 150, 186, 243, ? Let's identify the pattern governing this number series.

Analyzing the Number Series Pattern

We start by calculating the differences between consecutive terms in the series to see if there is a simple arithmetic progression.

The sequence of differences is 12, 19, 24, 38, 36, 57. This sequence does not immediately reveal a simple arithmetic or geometric pattern. Calculating the differences between these differences (second-level differences) also does not yield a clear pattern.

Identifying Sub-Series Patterns

Let's investigate further by splitting the original series into two separate sub-series: one containing terms at odd positions (1st, 3rd, 5th, 7th) and another containing terms at even positions (2nd, 4th, 6th, and the missing term).

Odd Positioned Terms Analysis

The terms at odd positions are 57, 88, 150, 243.

Calculating the differences between these terms:

• $88 - 57 = 31$
• $150 - 88 = 62$
• $243 - 150 = 93$

The differences are 31, 62, 93. This sequence forms an arithmetic progression, where each difference increases by 31. Specifically, the differences are $31 \times 1$, $31 \times 2$, $31 \times 3$.

Even Positioned Terms Pattern

The terms at even positions are 69, 112, 186. We need to find the next term in this sub-series, which corresponds to the missing number in the original series.

Calculating the differences between these terms:

• $112 - 69 = 43$
• $186 - 112 = 74$

The differences are 43, 74. Let's find the difference between these two differences:

• $74 - 43 = 31$

This indicates that the differences between the terms in the even-positioned sub-series also form an arithmetic progression with a common difference of 31.

Calculating the Missing Number in the Series

To find the missing number, we follow the pattern identified for the even-positioned terms.

The last difference calculated in the even sub-series was 74.

Based on the arithmetic progression of differences (with a common difference of 31), the next difference in this sequence should be:

$74 + 31 = 105$

Now, we add this calculated difference (105) to the last known term in the even-positioned sub-series (186) to find the missing number:

Missing Number = $186 + 105$

Missing Number = $291$

Thus, the number that replaces the question mark (?) in the given series is 291.