How can I get to know that the given sequence is in AGP? I know that a general term of AGP, $T_n = \{a + (n-1)d\}br^{n-1}$. Do I have to write the terms of a given sequence in this form to understand it is in AGP or not?
2026-03-26 04:50:52.1774500652
How can one identify a sequence as an Arithmetico-Geometric Sequence?
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It's not entirely clear to me just what you are looking for, but here's one possibly relevant fact:
A sequence $T_n$ has closed form of the form $Ar^n + Bnr^n$ (your formula is of this form once you expand and group similar terms) for some constants $A, B$ if and only if it satisfies the linear recurrence $T_n = 2rT_{n-1} - r^2T_{n-2}$. Depending on context, this may be an easier thing to check. Moreover it tells you how to determine $r$ from the first few terms of your sequence.