Problem: Investigate boundedness and monotonicity of

Solution: We try to calculate the first few terms: a₂ = 4, a₃ = 5/2, a₄ = 2, a₅ = 7/4. The first impression is that the sequence could be decreasing. What is the long term tendency? When n is large, the parts "+2" and "−1" can be ignored (see Intuitive calculations in Theory - Limit) and we get n/n = 1. So the sequence seems to go to 1, therefore it should be bounded (see the appropriate theorem in Theory - Limits - Basic properties). Also, since 1 is less than a₅ = 7/4, it seems possible that this sequence keeps going down. Thus we guess that it is decreasing.

Proof of boundedness:
First, clearly a_n > 0 for n ≥ 2. This shows that the sequence is bounded from below.

From our calculations above we guess that a_n ≤ 4. We will try to prove it. Note that since we assume n ≥ 2, we are multiplying the inequality below by a positive number; therefore we do not have to change the direction of this inequality (we will make a note about it on the right).

The last inequality is true for all n ≥ 2. Since the operations were equivalent, also the first inequality is true, exactly as needed.

Proof of monotonicity:
Now we will try to prove that the given sequence is decreasing. Again, note that we multiply out the inequality by positive numbers.

Since the last inequality is true and the operations were equivalent, the first inequality must also be true and the monotonicity is confirmed.

We can also try to use methods from theory of functions. We pass to investigation of functions (see Sequences and functions in Theory - Limits) and check on monotonicity of the function . We find the derivative:

Since this derivative is always negative, the function f is decreasing on (1,∞) and consequently also our sequence is decreasing.

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