Answer:
1. an = (-1)^(n-1)·(n+2)!/3^n
2. the sequence diverges
Step-by-step explanation:
Perhaps you're concerned with the sequence ...
{2, -24/9, 120/27, -720/81, ...}
1. This is neither arithmetic nor geometric. Ratios of terms are -4/3, -5/3, -6/3.
The alternating signs mean one factor of the general term is (-1)^(n-1). The divisors of 3 in the term ratios indicate 3^-n is another factor. The increasing multipliers suggest that a factorial is involved.
If we rewrite the sequence factoring out (-1)^(n-1)/3^n, we have ...
{6, 24, 120, 720, ...}
corresponding to 3!, 4!, 5!, 6!. This lets us conclude the remaining factor is (n+1)!.
The general term is ...
![\boxed{a_n=\dfrac{(-1)^{n-1}(n+2)!}{3^n}}](https://tex.z-dn.net/?f=%5Cboxed%7Ba_n%3D%5Cdfrac%7B%28-1%29%5E%7Bn-1%7D%28n%2B2%29%21%7D%7B3%5En%7D%7D)
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2. The magnitude of the factorial quickly outstrips the magnitude of the exponential denominator, so the terms keep getting larger and larger.
The sequence diverges.
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3. No series are provided.