Michael drove from home to work at an average speed of 45 mph. His return trip home took 45 minutes longer because he could only
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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 ...
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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.