Sn=(n^2+n)^1/2 - n
Prove that Sn < 1/2 for all n
Adding 1/4 to the term in parentheses must increase it, for all n. Therefore,
Sn < (n^2 + n + 1/4)^1/2 -n
= (n +1/2) -n = 1/2
Can you make it a bit longer please its supposed to be 15 marks
No. What is a mark, anyway? You should be able to elaborate the proof yourself.
mark is like grade. Also how is 1/2 < 1/2
I never said 1/2 < 1/2.
I started out by saying that, since
Sn = (n^2 + n)^1/2 - n, it must be less than (n^2 + n + 1/4)^1/2 + n.
The reason for that should be obvious.
To prove that Sn < 1/2 for all n, we can start by simplifying the expression for Sn.
Sn = (n^2 + n)^1/2 - n
Let's multiply and divide the expression by the conjugate of the numerator, which is (n^2 + n)^1/2 + n:
Sn = [(n^2 + n)^1/2 - n] * [(n^2 + n)^1/2 + n] / [(n^2 + n)^1/2 + n]
Using the difference of squares, we can simplify the numerator:
Sn = [(n^2 + n) - n^2] / [(n^2 + n)^1/2 + n]
Sn = n / [(n^2 + n)^1/2 + n]
Now, let's prove that Sn < 1/2 for all n by analyzing the expression.
1. First, notice that the denominator [(n^2 + n)^1/2 + n] is always positive because both n^2 + n and n are positive for all positive values of n.
2. In the numerator, we have n, which is always positive for positive values of n.
3. Since the numerator (n) is always less than the denominator [(n^2 + n)^1/2 + n] for positive integers n, we can conclude that Sn < 1 for all n.
Therefore, we have proved that Sn < 1/2 for all n.