Can anyone solve this mathematical induction problem?

[ctothe]

Homework Statement

Prove that for all n>1,
P(n) :1 + 1/2 + 1/3 +...+1/n = k/m

where k is an odd number an m is an even number.

Homework Equations

The Attempt at a Solution

1)Base Case n =2
P(2) = k/m
3/2 = k/m which is true.

2) Inductive Step
Assume P(n) is true for some arbitrary n.

3) Prove P(n+1)
P(n+1) = k/m
P(n) +1/(n+1) = k/m
We know/assume that P(n) has an odd numerator and an even denominator. So,
k/m + 1/(n+1) = k/m
(k(n+1) +m)/ (m+mn)

So i divided the problem into two cases:

Case 1: n+1 is odd
if n+1 is odd then k(n+1) + m is odd and m +mn is even which is true.

Case 2: n+1 is even
here's where the problem is

 

[rude man]

1. Case 2: n+1 is even
here's where the problem is

Let me rewrite your denominator mn + m = m(n+1).

Multiply m(n+1) by 2^p.

Show that there exists a positive integer value for p such that the numerator must wind up odd whereas the denominator will always be even.

 

[CEL]

Homework Statement

Prove that for all n>1,
P(n) :1 + 1/2 + 1/3 +...+1/n = k/m

where k is an odd number an m is an even number.

Homework Equations

The Attempt at a Solution

1)Base Case n =2
P(2) = k/m
3/2 = k/m which is true.

2) Inductive Step
Assume P(n) is true for some arbitrary n.

3) Prove P(n+1)
P(n+1) = k/m
P(n) +1/(n+1) = k/m
We know/assume that P(n) has an odd numerator and an even denominator. So,
k/m + 1/(n+1) = k/m
(k(n+1) +m)/ (m+mn)

So i divided the problem into two cases:

Case 1: n+1 is odd
if n+1 is odd then k(n+1) + m is odd and m +mn is even which is true.

Case 2: n+1 is even
here's where the problem is

k is odd. So, k(n+1) is odd, no matter if (n+1) is odd or even. Since m is even, k(n+1) + m is odd.

 

[rude man]

k is odd. So, k(n+1) is odd, no matter if (n+1) is odd or even. Since m is even, k(n+1) + m is odd.

Huh?

If k is odd and (n+1) is even, k(n+1) is even!

Odd x even = even always!