Induction question... (1 Viewer)

[dlesmond]

Hey I have a difficult (for me anyway) induction question that I can't seem to solve. Can someone please help me out? Any help would be appreciated...

I have typed up the steps that I can do so far...

Word file attached.

 

[vulgarfraction]

You're going about it the wrong way- you look as if you're trying to find n or something. The question's wrong, it should be greater than OR equal to (equality for n=1)

base case: true for n=1

inductive step:

suppose (x+y)^k-x^k-y^k >= 0

you then want to prove (x+y)^(k+1)-x^(k+1)-y^(k+1) >= 0


(x+y)^(k+1)-x^(k+1)-y^(k+1) = x[(x+y)^k-x^k] + y[(x+y)^k-y^k]
RHS > (x+y)[(x+y)^k-x^k-y^k] as x and y are positive
which is positive as both factors are positive, from your assumption

 

[dlesmond]

Cheers for your help, but I don't think that's what the question's asking for. It's a dumb question anyway.

 

[martinb]

Solution attached - hope it makes sense

 

[Chinmoku03]

I'm thinking whether there is a restriction on what values x and y can be, but eh

Prove that (x+y)ⁿ > xⁿ+yⁿ, n >= 1

Step 1: Prove true for n = 1
Sub n = 1:
Let LHS = x+y
Thus LHS >= RHS
Thus true for n = 1

Step 2: Assume true for n = k, k >= 1
So (x+y)^k > x^k+y^k

Step 3: Prove true for n = k+1, k+1 >= 1
Sub n = k+1
Let LHS = (x+y)^k+1
Thus LHS = (x+y)(x+y)^k
Thus LHS > (x+y)(x^k+y^k)
Thus LHS > x^k+1+y^k+1+x^ky+xy^k
Thus LHS > x^k+1+y^k+1
Thus LHS > RHS
Thus true for n = k+1

Therefore, (x+y)ⁿ > xⁿ+yⁿ, n >= 1

EDIT: Oops, forgot it said all positive integers, so 1 is included >.<; Fixed my answer up

 

[dlesmond]

Cheers for all your help, some dumb question hey...