Familiar number families can make a sequence easier to recognize. Squares, cubes and simple offsets from them are useful candidates, provided the relationship fits the full set of terms.
Consecutive squares
The sequence 1, 4, 9, 16, 25 follows 1², 2², 3², 4², 5². Continuing gives 6² = 36.
You can check the same structure using differences: 3, 5, 7 and 9. The next increase is 11. This provides a second route to 36 rather than relying on recognition alone.
Squares with an offset
For 2, 5, 10, 17, 26, subtract one from each term. The result is the square sequence 1, 4, 9, 16, 25. The next term under the rule n² + 1 is 37.
The offset must stay consistent. It is not enough that several numbers are close to squares if each requires a different adjustment.
Consecutive cubes
The sequence 1, 8, 27, 64 uses 1³, 2³, 3³ and 4³. The next cube is 5³ = 125. Cubes grow faster than squares, so a rough size check helps distinguish the two patterns.
For 0, 7, 26, 63, the terms are one less than those cubes. The next value would be 124.
Do not force a recognizable family
Seeing 16 and 25 does not prove that a sequence consists of squares. Check the order, spacing and every other term. A familiar number can appear within a completely different rule.
Practice the relationship, not a long list
Knowing a few small powers can help you generate a hypothesis quickly. Then verify it with arithmetic and the full pattern. If the relationship fails, move to differences, ratios or alternation rather than adding arbitrary exceptions to preserve an attractive first guess.