Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Translate an explicit geometric rule into a recursive rule
Problem
Translate the explicit geometric rule \(a_n~=~6(2)^{n~-~1}\) into a recursive rule.
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What this problem is really about
An explicit geometric formula contains both ingredients of its recursive form. We’ll evaluate at the first index to recover the initial term, identify the exponential base as the common ratio, and use that factor to relate each later term to its predecessor. Computing the second term both ways checks the starting index and multiplier.
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