Generate terms from recurrence and position rules

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Generate terms from recurrence and position rules

A term-to-term rule uses a preceding term; a position rule uses the term number directly. To generate a recurrence you also need enough starting information. The rule add 3 does not specify a unique sequence until its first term is stated.

For a position rule 3n + 2 with n starting at 1, the first five terms are 5, 8, 11, 14, 17. Substitute the position, not the previous term. Different rules can agree for a short initial list, so a few terms alone do not determine every future term.

Check indexing carefully. A sequence defined from n = 0 has a different first displayed term from the same formula defined from n = 1. State the convention and keep it through tables, examples and predictions.

Worked example: Generate the first three terms of 3n + 2 for n = 1, 2, 3.

Substitute each position.

Calculate multiplication before addition.

Answer: 5, 8, 11.

Check your understanding: Does “double the previous term” alone specify the first term?

Answer: No.

Why: A starting term is required.