Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Determine which term of an explicit sequence first meets a given condition
Problem
For the sequence rule \(a_n~=~10~+~5(n~-~1)\), determine when this condition is first met: \(a_n~=~40\).
Big Picture
What this problem is really about
To locate the term meeting a condition, turn the condition into an equation involving the explicit rule. We’ll set the term expression equal to the target, isolate the grouped transition count, and solve through the index shift without dropping it. Substitution and a neighboring-term check confirm both the value and that this is the first admissible index.
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