Question

I eliminated 2 of the 5 and now I have 3 I am confused towards which is the right one? help mathematical induction to determine which formula is true for all natural numbers n ≥ 3
a) 2n2 > (n + 1)2
b) (n − 1)2 > n2 − 6
c) (n + 1)2 < n2 +

Answer 1

a. Let n = 3.
2*3^2 > (3+1)^2.
18 > 16. True.

Let n = 4.
2*4^2 > (4+1)^2.
32 > 25. True.

Check b, and c.

Answer 2

(n+1)^2 = n^2+2n+1
So, check to see whether

2n^2 > n^2+2n+1
n^2 > 2n+1
Clearly, if n >= 3, n^2 >= 3n > 2n+1

(n-1)^2 > n^2 - 6
n^2 - 2n + 1 > n^2 - 6
2n - 1 < 6
2n < 7
Not true for n > 3

(c) is incomplete, but should be easy to check in similar ways.