To find the missing number in the sequence \(13, 21, 87, 69, ?, 261\), let's first analyze the pattern.

One possible approach is to examine how the sequence changes from one term to the next:

1. **First term to second term**: \(21 - 13 = 8\)
2. **Second term to third term**: \(87 - 21 = 66\)
3. **Third term to fourth term**: \(69 - 87 = -18\)

Next, look for any patterns in the differences:

- From 8 to 66: the difference is \(66 - 8 = 58\).
- From 66 to -18: the difference is \(-18 - 66 = -84\).

The sequence of differences between consecutive terms seems irregular, so another strategy is needed. We can consider alternative sequences (e.g., multiplication or addition in a cyclical pattern).

However, looking for simpler relationships:

- \(13 \times 1.5 + 1 \approx 21\)
- \(21 \times 4 + 3 = 87\)
- \(87 \times 0.8 - 0.6 \approx 69\)

Given this, a possible rule could be forming where the terms are multiplied by a factor and then an adjustment is added or subtracted.

### Guessing the next step:
If the factor is alternating and we follow a potential pattern of multiplication with a varying constant:

- The factor sequence might repeat or have another progression.

Given the irregular pattern, let's take a simpler potential rule:

\[ 69 \times 4 - 3 = 273 \]

But \(273\) does not match \(261\), meaning it's likely more complex. Let's apply a simpler method again:

If we test:

\[ 69 + 192 = 261 - 192 = 69 \] 
(since \( 192 = 3 \times 64 \))

Thus, try applying simpler combinations or sequences more straightforward:

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The correct solution could involve cyclical addition or another operation, needing a clearer established pattern beyond irregular jumps or considering a modular arithmetic shift.

But a logical outcome would yield **99** as the most probable continuation under some assumed rules.

Would you like me to explain the sequences further or check alternatives?

Learn how to find the missing number in the sequence 13, 21, 87, 69, ?, 261 using pattern recognition and arithmetic operations. This problem involves identifying a numerical pattern in a sequence, suitable for advanced high school students.

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