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?