I have been given that a1 = 1 and an+1 = 1/3*(an + 4). In order to prove that this sequence is monotonous, what is the second step of mathematical induction? If my explaining of the question is unclear, here is a picture of the question and the possible answers. I believe the answer is C but I welcome any correction/confirmation.
gyazo . com/85e9ff97c6139e1d0ccb1054f1c30e5e.png
Monotonic*
you need to show that
a<n+1> > a<n> (increasing)
or
a<n+1> < a<n> (decreasing)
If you expand the expression from the recursion, you can clearly see that it is.
Once you can show that each term is greater (or less) than the previous one, then you have shown that the entire sequence is monotonic (and thus, monotonous!).
Thank you Steve for the response. So from what you have said, I first assume that a<n+1> > a<n> and then prove that the corresponding expression is true? Or is it the other way around?
To prove that a sequence is monotonous using mathematical induction, we need to show that a1 ≤ a2 and that an ≤ an+1 for all positive integers n.
In this case, the given sequence is defined as a1 = 1 and an+1 = 1/3*(an + 4). To prove that the sequence is monotonous, we first need to check whether a1 ≤ a2:
Substitute n=1 into the formula for an+1:
a2 = 1/3*(a1 + 4) [Using the given formula, replacing n with 1]
Substitute the given value for a1:
a2 = 1/3*(1 + 4) [Using a1 = 1]
Evaluate the right-hand side:
a2 = 1/3*(5)
a2 = 5/3
Now, we need to determine whether a1 ≤ a2 by comparing their values:
a1 = 1
a2 = 5/3
Since 1 ≤ 5/3, we have shown that a1 ≤ a2. This is the first step of mathematical induction to prove that a sequence is monotonous.
Now, let's move on to the second step of mathematical induction. According to the options given:
A) Show that an+1 ≤ an for all positive integers n.
B) Show that an < an+1 for all positive integers n.
C) Show that an ≤ an+2 for all positive integers n.
D) Show that an+1 ≤ an+2 for all positive integers n.
Based on the definition of a monotonous sequence, we need to show that an ≤ an+1 for all positive integers n. Therefore, the correct second step of mathematical induction for proving the given sequence is monotonous is option B:
B) Show that an < an+1 for all positive integers n.
So, your answer is not C but B.