What is the next number in this pattern?Rule: multiply by 2, then add 52,   9,   23,   51,   ...

1. Break Down the Problem

We need to determine the next number in the sequence by applying the given rule: multiply by 2, then add 52.

2. Relevant Concepts

The sequence is constructed by taking each term, multiplying it by 2, and then adding 52 to generate the next term.

3. Analysis and Detail

Let's analyze the existing numbers in the sequence:

• The first term is $9$.

• The second term can be calculated as: $9 \times 2 + 52 = 18 + 52 = 70$

• The third term is $23$.

• The fourth term can be calculated as: $23 \times 2 + 52 = 46 + 52 = 98$

• We observe that the term $51$ does not follow the same logic. Let's check:

  If we take the last valid term from the conversion sequence from $70$ or $98$ to $51$, it seems there may have been a misalignment or different logical sequence to connect between them.

We need to check and correct if reverting back to previous terms could yield clarity among them.

• To check for moving from previous $51$:
• From $51$: $? \times 2 + 52 = 51 \rightarrow ? = (51 - 52) / 2 = -0.5$

As it stands, it seems $51$ is an anomaly, but maintaining the sequence found initially.

4. Verify and Summarize

Since the sequence diverges at $51$, we can logically presume we stick with $23$ leading as base.

• To continue from the last valid term, continuing from more clear logical allowances:
  • Next applying to previous pace at 23: $51 \times 2 + 52 = 102 + 52 = 154$

Final Answer

The next number in the modified pattern should logically be $154$.