Write arithmetic and geometric sequences explicitly and recursively; translate between forms and model situations.
Compare two sequence representations at a specific term number
Problem
Compare these two sequence representations at \(n~=~6\): \(a_n~=~5~+~2(n~-~1)\) and \(b_1~=~4\); \(b_n~=~b_{n-1}~+~3\).
Big Picture
What this problem is really about
Two sequence representations can be compared only after both are evaluated at the same index. We’ll substitute the index directly into the explicit rule, count the one-fewer transitions needed by the recurrence from term one, and compute both values independently. Their signed difference then identifies which sequence is larger and by how much.
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