Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Translate a recursive geometric rule into an explicit rule
Problem
Translate the recursive geometric rule \(a_1~=~5\); \(a_n~=~2a_{n-1}\) into an explicit rule.
Big Picture
What this problem is really about
A geometric recurrence becomes explicit by counting repeated applications of its multiplier. We’ll extract the starting term and common ratio, observe that the first term uses zero extra factors, and express the later factor count with the index. Generating a term recursively and directly provides a clean equivalence check.
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