Answer:
a) To determine if the sequence {an} is monotone, we need to check if it is either increasing or decreasing.
Let's calculate the values of the first few terms of the sequence to observe any patterns:
a₁ = 4(1) / (4(1) - 1) = 4/3 ≈ 1.333
a₂ = 4(2) / (4(2) - 1) = 8/7 ≈ 1.143
a₃ = 4(3) / (4(3) - 1) = 12/11 ≈ 1.091
From the calculated values, we can see that the sequence is decreasing. As the value of n increases, the terms of the sequence decrease.
Therefore, the sequence {an} is monotone decreasing.
b) To show that the sequence {an} converges, we need to prove that it approaches a finite limit as n approaches infinity.
Let's rewrite the expression for the nth term of the sequence {an}:
an = 4n / (4n - 1)
As n approaches infinity, the term 4n-1 in the denominator becomes negligible compared to 4n.
Therefore, as n gets larger and larger, the value of an approaches 4.
We can demonstrate this by taking the limit of the sequence as n approaches infinity:
lim(n→∞) (4n / (4n - 1)) = 4
Since the limit of the sequence is a finite number (4), we can conclude that the sequence {an} converges.