Use the inductive method of proof to prove....?

Question

1. Prove the statement is true using mathematical induction:

1 + 3 + 5 + 7 + .... + (2n - 1) = n^2

2. Prove the statement is true using mathematical induction:

2n-1 ≤ n!

Answer 1

1) 1 + 3 + 5 + 7 + .... + (2n - 1) = n^2
for n = 1, its ok.
suppose it is proven until n = N-1
now we need to prove it for n = N

1 + 3 + 5 + 7 + .... + (2(N-1) - 1) + (2N-1) = (N-1)^2 + (2N-1)

first part until n = N-1 by induction hypotheses,

(N-1)^2 + (2N-1) = N^2 +1-2N + 2N -1 = N^2 yes !

proof finished.


2)
2n-1 ≤ n!

for n = 1 its ok. but not for n = 2,
so for n = 3 ok
n = 4 ok
for n>=3 :
n! = n* (n-1)! >= n * { 2(n-1) -1} = n {2n -3}

Answer 2

right this is the straightforward theory, you ought to fill in some wording. n=a million a million=a million^2 is real assume real for n, teach real for n+a million a million+3+...+(2(n+a million)-a million)= [a million+3+...+2n-a million]+(2(n+a million)-a million)= n^2+2n+a million= (n+a million)^2