Read a sequence rule

The rule takes a position as input

For the rule uₙ = 2n + 1, positions n = 1,2,3,4 give term values 3,5,7,9. The subscript n labels the position; it is not a number to add to the term after evaluating it.

The tenth term is 2 × 10 + 1 = 21. For vₙ = n + 3, the tenth term is 13 instead. Read the provided rule rather than continuing a different familiar list.

Check one position at a time

A table with positions 1,2,3 and values 4,5,6 fits the given rule vₙ = n + 3. At n = 1, the output is 4; at n = 3, it is 6. The first position is 1 unless the question states a different starting index.

Check your understanding 1 Substitute the position

For uₙ = 2n + 1, find u₆ and u₁₅. Explain why u₆ is not 6.

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The values are 13 and 31. Six is the input position in the first calculation, while 13 is the output term. A subscript identifies which term to evaluate.

A stated rule supplies the continuation

A short finite list alone can fit several rules. When the formula is given, use it for the requested positions. Finding a rule for an unknown pattern is a separate task.