EXT 1 Math - Mathematical Induction (help)

[fishbiscuit]

I'm having trouble understanding mathematical induction when inequalities are involved, especially in the 'prove for n = k + 1' part of the steps.
The question I'm doing is:

n^2 > n - 5 for all positive integers n

I've tested, assumed n = k and now trying to prove true for n = k + 1, but I don't know how to make the LHS = RHS, or if I even have to make them equal because it's an inequality.
I thought of subbing a random value in for k and testing it like that, but I'm not sure that's a reliable way of proving it because it would only be true for that particular value.
I'm just really confused on how inequalities work for this and would really like some help and clarification. Thank you.

[RuiAce]

Note: The statement \(3k > k\) is always true provided \( k > 0\), which we satisfy here from establishing that \(k\ge 1\).

[fishbiscuit]

Hello, thank you for helping!
I don't understand the substitution of k^2 to (k - 4) in the second line of your working out in the substitution section. How does k^2 = k - 4?

[RuiAce]

Hello, thank you for helping!
I don't understand the substitution of k^2 to (k - 4) in the second line of your working out in the substitution section. How does k^2 = k - 4?

Deleted an earlier reply - I made a typo in substitution; fixing it now

[fishbiscuit]

Ah, okay, that makes sense.
So, I can substitute what I assumed into my proof, and the sign I use is the one that is associated with what I sub in? (When you subbed in k - 5 for k^2 you used the '>' sign)

[RuiAce]

Ah, okay, that makes sense.
So, I can substitute what I assumed into my proof, and the sign I use is the one that is associated with what I sub in? (When you subbed in k - 5 for k^2 you used the '>' sign)

Basically, yep.

(Reason being the '>' and '=' signs, when nothing is written on the left, always refers to the thing above it.)

[fishbiscuit]

Thank you so much!!